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LIBRARY OF CONGRESS. 






UNITED STATES OF AMERICA. 



• \ 




TELESCOPE COMPASS. 



A MANUAL 



PLANE SURVEYING; 



CONFINED TO 



WORK WITH THE COMPASS. 



WITH AN APPENDIX. 



AMPLY ILLUSTRATED. 



BY |/ 

THOMAS BAGOT, 

SUPERINTENDENT OF RIPLEY COUNTY, INDIANA. 






INDIANAPOLIS, INDIANA^^V 

THE NORMAL PUBLISHING ' 



J. E. Sherbill, Proprietor. 

1883. 



WV 5 



Entered according to Act of Congress in the year 1881, 

By J. E. SHERKILL, 

In the office of the Librarian of Congress at Washington. 



CARLON & HOLLENBECK, WANAMAKER & CARSON, 

PRINTERS AND BINDERS, INDIANAPOLIS ELECTROTYPE FOUNDRY, 

INDIANAPOLIS. INDIANAPOLIS. 



30r&1?Ct 



PREFACE. 



Every person who studies Surveying from the text-books in gen- 
eral use, and afterward is called upon to discharge the duties of a 
surveyor, must, in the course of time, become aware of two things i 
(1) that he has spent time in learning much that he has never 
had, and probably never will have, occasion to use, and (2) that a 
great deal he needs to know, and must know, is not to be found in; 
the books. 

This is the case particularly under the Rectangular System, and 
the author's experience has led him to believe that a necessity ex- 
ists for a book dealing directly with the problems continually 
coming up before surveyors throughout the country, and that such 
a book will be cordially received by every person who wishes k> 
understand the subject as it is comprehended in general practice. 

And this, dear reader, accounts for the existence of this little 
book. You will find it simply a brief treatise on Compass-Sur- 
veying, shorn of everything superfluous, and yet embracing alf 
that is necessary to a good understanding of the subject. Very 
few geometrical or trigonometrical terms are employed, and alF 
the problems may be mastered by any person having a moderately^ 
good knowledge of arithmetic. 

The author does not claim that the book is above criticism, butt- 
on the contrary, he is well aware of the fact that a person disposed 
to criticise may find in it an ample field in which to exercise his 

(3) 



4 PREFACE. 

talents. He trusts, however, that a search for its faults will re- 
sult in disclosing enough merit, even among so much demerit, to 
excuse him for writing it, and so trusting, he submits it to the 
public. 

The plates of surveying instruments used in the book represent 
the excellent instruments manufactured by T. F. Eandolph, Cin- 
cinnati, Ohio. 

New 7 Marion, Indiana, 
May, 1883. 



CONTENTS 



CHAPTER I.— Introduction. 
No. of Art. 

( 1). Surveying defined. 
( 2). Branches. 

1. Topographical surveying. 

2. Geodetic surveying. 

3. Plane surveying. 

( 3). Measurements, how made. 

1. Actual area nearly always greater than computed area, 

2. Smooth surface conceived underneath. 

3. Impossible in many instances to compute the area of 

real surface. 
( 4). Corners defined. 
( 5). Surveying instruments used. 
( 6). Transit and compass. 

1. Eemarks on the transit. 

2. Eemarks on the compass. * 
(7). Chain and pins. 

( 8). Chaining over hills. 

( 9). Flag-staff, drawing instruments, etc. 

(10). Assistants needed. 

CHAPTER II. — Description of the Compass. 

(11). The compass circle, how divided. 

(12). Magnetic needle and center pin. 

(13). Degrees, how marked. 0° and 90° points. 

(14). Compass box, plate, sights, etc. 

(5) 



j6 contents. 

(15). Description of needle. Delicacy, how determined. 

(16). Horizontal angles, how measured. 

(17). Letters "E" and "W" reversed on compass face. 

(18). Eeason for this. 

(19). Rule for reading bearings. 

(20). Reverse bearing. 

(21). Northerly and southerly bearings. 

(22). Running lines east or west. 

(23). Rules for measuring angles. 

1. When both readings are in the same quadrant. 

2. When one reading is in each of either the two north 

quadrants or the two south quadrants. 

3. When one reading is in each of either the two east quad- 

rants or the two west quadrants. 

4. When one bearing is in each of two opposite quadrants. 
(24). Reasons for these rules. 

(25). Glass cover to compass box. Electricity, how excited in it, 

and how removed. 
(26). Needle affected in other ways. 
(27). Sight compass and telescope compass. Either may be either 

a plain compass or a vernier compass. 
•{"28). Description of the vernier. 
(29). How used. 
(30). Other kinds of verniers. 
(31). Repairing the compass. 

1. To re-magnetize the needle. 

2. To sharpen the center pin. 

3. To replace a spirit level. 

4. To adjust a new sight. 

5. To straighten the center pin. 

6. To straighten the needle. 

7. To put in a new glass. 

8. To regulate the movement of the ball. 
(32). The compass, how carried. 

(33). Needle should be lifted from center pin when the compass is 

not in use. 
(34) Compass should be kept level when not in use, and the needle 

allowed to assume its natural position. 
(35). The telescope compass gradually growing in favor. 



CONTENTS. / 

CHAPTER III.— The Variation of the Magnetic Needle. 

(36). Meridian defined. 

(37). The true meridian. 

(38). Methods of determining the true meridian. 

1. By a shadow. 

2. By Polaris. 

(a). Polaris and Alioth. 

(b). Pole between them. 

(c). The plumb-line. 

(d). Greater accuracy. 

(e). When upper culmination occurs during the day. 
(39). Table showing time of culmination of Polaris. 
(40). Magnetic meridian. 
(41). East variation and west variation. 
(42). Agonic line. 

(43). Isogenic lines. [variation of the needle. 

(44). The north magnetic pole; the effect of its movement on the 
(45). Secular change of variation. 
(46). Diurnal change and annual change. 
(47). Table showing diurnal change by hours. 
(48). Diurnal and annual changes usually disregarded in practice. 
(49). Electric disturbances. 
(50). The "dip" or inclination of the needle. 
(51). Many magnetic phenomena imperfectly understood. 

CHAPTER IV. — Effect of Change of Variation on Old 
Lines, and Methods of Correcting Bearings. 

(52). Bearings of lines subject to constant change, 

<53). Illustration. 

(54). Effect of re-surveying lines without considering the change. 

(55). Area of tract not affected. 

{56). To determine the present bearing of a line. 

1. Bearing at time of a previous survey. 

2. Date of previous survey. 

3. Annual amount of secular change. This may be de- 

termined in various ways. 
(1). By comparison of bearings. 
(2). By establishing a true meridian. 
(3). By interpolation. 



8 CONTENTS. 

(57). Table showing the variation of the needle at important sta- 
tions. 

(58). Table showing annual amount of secular change in certain 
localities. 

(59). Tables to be used in approximation. 

(60). Determining variation for past times. 

(61). Advantage of basing bearings on the true meridian. 

(62). Magnetic meridian bearings subject to constant change. 

(63). To determine the true bearing of a line from its magnetic 
bearing. 

1. When the variation is west. 

2. When the variation is east. 
(64). Bearings to be changed. 

(65). When supplement is to be taken. 

(66). Changing from the true meridian to the magnetic meridian. 

CHAPTER V. — Method of Running Lines. 

(67). Finding a corner, etc. 

(68). Determining position of corner from witnesses. 

(69). Where trees are not available for witnesses, other things are 
used. 

(70). Method of describing witness trees. 

(71). Starting the survey of a line. 

(72). Course and distance of line must at least be known approxi- 
mately before the line can be surveyed. 

(73). Setting the compass and measuring the line. 

(74). Setting off variation on the vernier. 

1. When the variation is east. 

2. When the variation is west. 
(75). Surveying the line. 

(76). How chained. 

(77). Pins, stakes, etc. 

(78). Relative places on line of men engaged in the survey. 

(79). Continuation of the line to the opposite corner. 

(80). In case the line does not strike the corner. 

(81). Illustration. 

(82). Rules for correcting the stakes. 

(83). Terminations of lines and starting point regarded as vertices 
of an isosceles triangle. 



CONTENTS. y 

(84). Examples in moving (correcting) the stakes. 

(85). Abbreviations used. 

(86). Errors caused by imperfection of instruments, etc. 

(87). Correction of assumed bearing. 

(88). Rules for correcting bearings. 

1st method* Derivation of rule. 

(1). Trigonometrical lines. 

(2). Sine and cosine defined. 

(3). Study of relation of sine and cosine important. 

(4). Application of these lines. 

(5). When the angle is large. 
2d method. 

(1). Modification of first method. 

(2). Illustration. 
3d method. When the bearing of the line at a previous 

time is known. 
(89). Examples of lines whose bearings are to be corrected. 
(90). Is amount of correction to be added or subtracted ? 

1. Rule for north-east and south-west courses. 

2. Rule for north-west and south-east courses. 
(91). Examples under these rules. 

(92). Thus far, all bearings have been based on the true meridian. 
(93). Bearings of lines based on the magnetic meridian. 

1. Rule for correction. 

2. Demonstration of the rule. 

(94). Rule good while the needle moves westward. 

(95). Bearings to be corrected. 

(96). Abbreviations, etc. 

(97). Completion of the survey. 

(98). Assistants generally sworn. 

(99). Instruments should be tested frequently. 

(100). Backsights. 

CHAPTER VI. — United States Rectangular Surveying* 

(101). Public lands, how divided. 

(102). Townships, how divided. 

(103). These provisions sufficient. ' 

(104). Fundamental lines and initial point. 

(105). Some imperishable mark chosen for the initial point. 



10 CONTENTS. 

(106). Survey of range lines or meridians, etc. 

(107). Survey of parallels, etc. 

(108). Convergence of meridians. Correction lines. 

(109). Auxiliary meridians, etc. 

(110). Survey of townships north of the base-line and east of the 

principal meridian. 
(111). Survey of townships north of the base-line and west of the 

principal meridian, etc. 
(112). Excesses and deficiencies. 
(113). Congressional township and civil township. 
(114). Survey of sections. 

1. Preliminaries. 

2. How to obtain bearings. 

3. Sections, how numbered. 

4. Survey of section 36. 

5. Survey of the rest of eastern tier of sections. 

6. Completion of the township. 

(115). Full sections and fractional sections. The " double frac- 
tional." 

(116). Meander corners and meander lines. 

(117). Making up the field-notes. 

(118). Monuments adapted to the country surveyed. 

(119). Work of Government deputy extends only to the division of 
the township into sections. 

(120). Advantages of the Rectangular system, etc. 

CHAPTER VII. — The Division and Subdivision of the 

Section. 

(121). Subsequent surveys must be made in accordance with the 

original. 
{122). The ideal section and the real section. 
U23). The divisions and principal sub-divisions of the section. 
(124). Names of these divisions and sub-divisions. • 
(125). Descriptions of land generally qualified by the words " more 

or less." 
(126). Corners, how named. 

1. Section corners. 

2. Quarter-section corners. 

3. Half-quarter corners. 

4. Fourth-quarter corners. 



CONTENTS. 11 

(127). Section lines and center lines. 
(128). Position of corners, how determined. 

1. Section corners and exterior quarter-section corners. 

2. Center corners, methods of setting. 

(1). By crossing the center lines. 

(2). By bisecting east and west center line. 

3. Half-quarter corners. 

4. Fourth-quarter corners. 

5. Other corners, 

(129). Examples in setting corners. 

(130). Survey of the divisions and sub-divisions of the section. 

1. Quarter section. 

2. Half-quarter section. 

3. Fourth-quarter section. 
(131). General rule. 

(132). When some of the boundary lines are known. 

(133). Tracts to be surveyed. 

(134). Independent corners, lines, and tracts. 

(135). Illustration of independent corners, etc. 

(136). Dependent corners, lines, and tracts. 

(137). Dependent lines, etc., how surveyed. 

(138). Examples of dependent tracts. 

(139). Description by "metes and bounds." 

(140). Rules for setting corners, etc., in full sections, generally 

apply to fractional sections. 
(141). When a tract of land lies partly in one section and partly 

in another. 

CHAPTER VIII.— Field-Notes. 

(142). Field-notes of sectional survey by the Government deputy. 
(143). Contents of full section, supposition regarding it. etc. 
(144). Plot of township. 
(145). Explanation of the plot. 
(146). List of witnesses to corners. 

1. Exterior corners. 

2. Interior corners. 
(147). Other particulars. 

1. Area of fractional quarters. 

2. Creeks, etc. 

3. Offsets. 



12 CONTENTS. 

(148). Surveyor needs copy of original field-notes. 
(149). What surveyors' records should contain. 
(150). Pocket record. 
(151). Description of pocket record. 

1. Left-hand page. 

2. Eight-hand page. 

(152). Instructions regarding bearings. 
(153). Kepresentation of surveys on the plot. 
(154). Approximating the bearing of a line. 
(155). Illustration. 

(156). Principles underlying the method. 
(157). The field-book. 

(158). Method of keeping the field-book in independent surveys. 
(159). When new witnesses should be taken. 
(160). Method of keeping the field-book in dependent surveys. 
(161). Names of stations, witnesses, etc. 

(162). Kecords by authorized surveyors taken as prima facie evi- 
dence in favor of surveys recorded. 
(163). Other methods of keeping field-notes. 

CHAPTER IX.— Re-location of Corners. 

(164). Trouble caused by lost corners. 

(165). Nature of this trouble. Means of re-locating corners. 

1. Remains of missing corner or witnesses. 

2. By course and distance from some other corner. 

3. By retracing old line by marks on trees, etc. 

4. By projecting lines. 

(1). Illustration of this method. 

(2). The reverse of the method by which the corners 

were established. 
(3). Another illustration. 
(4). Examples for practice. 

5. A last resort. 

(1). Setting S. ] corner. 

(2). May not agree in position with corner lost. 
(3). A case illustrated. 
(4). Lost section corner. 
(5). Case of disagreement illustrated. 
% (6). When a quarter corner can not be found. 



CONTENTS. 13 

(166). Re-locating original corners to the variable quarters of 

fractional sections. 
(167). To set a quarter corner between two fractional sections. 
(168). To set an exterior corner to a fractional section, or to any 

exterior section. 

1. When there is an offset. 

2. When there is no offset. 
(169). Examples. 

(170). Re-location of subsequent corners, etc. 

CHAPTER X.— Descriptions of Land. 

(171). Necessity of a good description. 

(172).. Length of lines and area of tracts should be given in sur- 
veyor's measure. 

(173). Tables of equivalents. 

(174). Examples for reduction. 

(175). Fractions of a chain expressed in links. 

(176). General rule for reduction. 

(177). Area of tracts, how expressed. 

(178). Description of independent tract. 

(179). Examples of errors in descriptions. 

(180). Examples for correction. 

(181). Erroneous descriptions. 

(182). Use the words " more or less." 

(183). In describing dependent tracts the course and distance of 
each boundary should generally be given. 

(184). The description should state whether the bearings are based 
on the true meridian or on the magnetic meridian. 

(185). Lines running north, etc. 

(186). Surveys generally made in accordance with the description. 

CHAPTER XL — Obstacles to Alignment and Measurement. 

(187). Obstacles met on line. 

(188). Two classes of obstacles. 

(189). Methods of spanning obstacles of first class. 

1. By perpendiculars. 

2. By an equilateral triangle. 

3. By a right-angled triangle. [urement. 

(1). When an obstacle both to alignment affd meas- 
(2). When an obstacle to measurement alone. 



14 CONTENTS. 

4. By symmetrical triangles. 

5. When a fence is built on or near the line. 

(1). When offset line terminates on the opposite side. 

(2). When the offset line terminates on the same side. 

(a). When the distance missed is greater than 

the offset. 
(b). When the offset is greater than the distance 
missed. 

6. Surveying over hills. 

(190). Methods of spanning obstacles of the second class. 

1. By a right-angled triangle. 

2. By symmetrical triangles. 

3. By similar triangles. 
(191). Other methods, etc. 

CHAPTEE XII.— Computation of Area. 

(192). Advantage of expressing dimensions of tracts in chains and 

links. 
(193). Special rules deduced. 
(194). Examples in computation. 
(195). Every tract of land a polygon in shape. 
(196). Bectangles. 
(197). Parallelograms. 
(198). Trapezoids. 
(199). Triangles. 

1. With base and altitude given. 

2. With no altitude given. 
(200). Trapeziums. 

(201). Any figure. 

(202). Computation of area by latitudes and departures. 

(203). Latitude and departure, as applied to courses, defined. 

(204). North and south latitudes, and east and west longitudes. 

(205). Signs of latitudes and departures. 

(206). Relation of sine and cosine to departure and latitude. 

(207). Table of natural sines and cosines explained. 

(208). Examples in computing latitudes and departures. 

(209). Columns marked differently at top and bottom. 

(210). Traverse tables. 



Contents, 15 

(211). Sum of latitudes and sum of departures in every cor- 
rect survey. . 



(212 
(213 
(214 
(215 
(216 
(217 
(218 
(219 
(220 
(221 
(222 
(223 
(224 
(225 
(226 
(227 
(228 



A trial survey. 
Explanation. 

When a discrepancy exists. 
Initial line or meridian. 

Difference between longitudes and departures. 
How to determine the longitude of a course. 
Algebraic sum must be used. 
Simplification of the rule. 
Computation of area by longitudes. 
North products and south products. 
Double longitudes. 
Method of keeping the data. 

General rule for computing areas by double longitudes. 
To determine the most westerly corner. 
Examples in computation. 
Courses without departures, etc. 

Not absolutely necessary that the meridian be drawn, 
through the most westerly station. 



CHAPTER XIII.— Laying Out and Dividing Up Land. 

(229). No general rule can be given. 

(230). What are known in problems to be considered. 

(231). To lay out a square. 

(232). To lay out a rectangle. 

(233). To lay out a parallelogram. 

(234). To lay out a right-angled triangle. 

(235). To lay out a trapezoid. 

(236). To lay off any figure. 

(237). Things to be considered in making partition. 

(238). Nature of problems chosen. 

(239). To divide a rectangle into equal parts. 

(240). To divide a rectangle into unequal parts. 

(241). Problems. 

CHAPTER XIV.— Surveying Town Lots. 

(242). Description of town lots, blocks, etc. 
(243). Survey of town, how based. 



16 CONTENTS. 

(244). Usual shajje of lots, etc. 

(245). What the plot of- a town should show. 

(246). Particulars respecting Figure 58. 

(247). Illustration of a town in which the lots vary in size. 

(248). Method of surveying lot number 11 in the figure. 

(249). Examples for practice. 

(250). Chain or tape used in surveying should be tested frequently. 

CHAPTER XV.— Plotting. 

(251). Plotting denned. 
(252). Instruments used. 

1. Drawing board. 

2. T-square. 

3. Ruler. 

4. Drawing pen. 

5. Dividers or compasses. 

6. Pi o tractor. 

7. Diagonal scale of equal parts. 

(253). Units of the scale may have various equivalents. 

(254). Plotting bearings. 

(255). Examples for practice. 

(256). Plotting rectangular tracts. 

(257). Plotting tracts in general. 

(258). A particular case. When a survey does not " close." 

(259). The pantograph. 

(260). Locating objects on the plot. 

(261). Coloring plots and maps. 

CHAPTER XVI.— Surveying Without a Compass. 

(262). The compass nearly always necessary. 

(263). Setting corners. 

(264). Establishing lines. 

(265). Setting out perpendiculars. 

(266). Survey of rectangular tracts. 

(267). Measurements. 

APPENDIX. 

Land Decisions. 

Table of natural sines and cosines. 



MANUAL 



OF 



PLANE SURVEYING. 



CHAPTER I. 

* INTRODUCTION. 

Art. ( 1 ). Surveying is that branch of applied mathematics 
which embraces operations for finding, (1) the relative positions 
of points on the earth's surace, (2) the area of any portion of its 
surface, and (3) the contour or shape of any part of its surface, so 
that it may be represented in maps and plots. 

( 2 ). It is divided into three branches : 

1. Topographical Surveying, or Topography, includes operations for 
determining the contour of portions of the earth's surface and re- 
presenting it on paper. 

2. Geodetic Surveying, or Geodesy, takes into consideration the 
curvature of the earth's surface and is employed in extensive sur- 
veys. 

3. Pkme Surveying does not regard the curvature" of the earth's 
surface and all lines are measured as on a plane. It is used in lo- 
cal work. 

(3). All measurements in surveying are made as nearly hori- 
zontally as possible, and the area of a tract of land is not its ac- 
tual surface measure, unless the tract be perfectly level, but the 
amount of land enclosed by its boundaries measured horizontally, 
instead of with the inclinations of the surface over which they run. 
2 (17) 



18 MANUAL O^ PLANE SURVEYING. 

1. The actual area," therefore, is nearly always greater than its 
computed area, and increases in proportion to the inequality of 
the surface. 

2. We may conceive a smooth surface at the level of the ocean 
underlying the surface of the ]and; then the area of a tract of 
land is equal to the contents of a figure formed by projecting the 
boundaries of the tract on the horizontal surface below. 

3. Were the real surface considered, it would be impossible in 
many instances either to compute its area or represent its contour 
on paper. 

(4). The extremities of lines in surveying are called corners, 
and each corner marks the vertex of an angle formed by the meet- 
ing of two lines. The corner is a mathematical point, and may 
or may not be marked with a monument. 

. ( 5). Lines are surveyed either with a solar or a magnetic in- 
strument. With the former, where more precision than expedi- 
tion is required ; and with the latter, where expedition is of more 
importance than precision. 

( 6 ). The principal magnetic instruments in use are the transit 
and compass. 

1. The transit is provided with a telescope, and at the present 
time is so constructed as to be adapted to the measurement of 
both horizontal and vertical angles. It serves many important 
purposes independent of the assistance of the magnetic needle, and 
is not strictly a magnetic instrument. 

2o The compass is either supplied with sights or a telescope^ 
and is strictly a magnetic instrument. It is not usually adapted 
to measuring vertical angles. The lightness, simplicity, arid con- 
venience of the compass have brought it into almost general use 
in common surveying, and the following chapter is devoted to a 
description of it. The transit may be found described in almost 
any comprehensive work on Surveying. 

( 7 ). Measurements of lines in surveying are made with an 
iron or steel chain* usually 33. feet or 2 rods long and divided 
into 50 links. It has a handle at each end by which it is carried 
during the survey, and the successive chain-lengths are maked 

*For convenience in counting the links, this chain is divided into five 
parts of ten links each by four brass or copper tags. The real chain is 100 
links or 66 feet in length, 'and called a Gunter's chain, from its inventor. 
The half-chain of 50 links is used for convenience. A link is 7.92 inches in 
length. 



MANUAL OF PLANE SURVEYING. 19 

with pins of iron or steel wire, generally 10 or 12 inches in length, 
sharpened at one end and bent into the form of a ring at the other. 

In the ring is soruetimes'tied a bright ribbon or piece of cloth to 
render the pin more conspicuous, in order that it may be easily 
found by the person who carries the rear end of the chain, and 
such colors should always be chosen as will contrast most with 
the surface to be surveyed. 

( 8 ). In chaining up and down hill, the chain must be kept taut 
and horizontal, as on a level surface. In order to do this, it is 
sometimes necessary to drop the pin from the front end of the chain, 
or elevate the rear end of the chain to a point exactly over the 
pin that sticks in the ground. 

(9). A straight staff, about 1J inches in diameter and 8 or 10 
feet high, surmounted by a small flag of brilliant color, is used in 
alignment ; and a good set of drawing instruments, drawing board? 
t-square, triangle, protractor, ruler, etc., are necessary in drawing 
and plotting. 

(10). In field-work the surveyor generally needs four assist* 
ants ; two chain-men, one flag-man, and a marker. The first two 
measure the line with the chain, the third carries the flag, and 
the fourth assists the chain-men by marking the line with stakes 
at the proper distances. To this force it is sometimes necessary 
to add one or more ax-men, as bushes may have to be cut out of 
the way and trees marked. 

Questions on Chapter I. 

1. Define Surveying. 

2. Into what three branches is it divided ? 

3. What does each branch embrace ? 

4. How are measurements made in surveying? 

5. Why is the actual area of a tract of land nearly always 

greater than its computed area? 

6. What is a corner? 

7. Name the two kinds of instruments used in surveying. 

8. Name the principal magnetic instruments. 

9. Describe the transit. The compass. 

10. How are lines measured in surveying? 

11. How should the chain be held in chaining over hills? 

12. How many assistants does the surveyor usually need? 



CHAPTEE II. 

DESCRIPTION OF THE COMPASS. 

(11). The compass consists essentially of the compass circle, 
the magnetic needle, and the sights. The circle has its circum- 
ference raised and divided into 360 equal parts or degrees, and 
these are usually subdivided to half or quarter degrees. 

( 12 ). At the center of the compass circle is placed a perpen- 
dicular pin, called the center pin. Upon this pin the magnetic 
needle is balanced in such a way as to mark opposite points of the 
divided circumference of the circle. 

(13). The degrees on the circumference are marked from two 
opposite points, called points, up to 90° to the right and left. 
The 90° points are, therefore, opposite one another also. One of 
the points is called the north point, and the other the south 
point, and one of the 90° points is called the east point, and the 
other the west jDoint. 

( 14). The compass circle is enclosed in what is known as the 
compass box, and this rests upon the compass plate. The box is 
usually between 5 and 7 inches in diameter, and the plate about 
12 or 14 inches long. At the ends of the plate perpendicular 
sights are placed. These sights have slits in them, and are so 
placed that the line of sight from one of them to the other will 
strike opposite points of the graduated circumference of the com- 
pass circle. Between the compass box and the sights are usually 
placed two spirit levels, one at right angles to the other. These 
are used in leveling the compass. The compass rests upon a tri- 
pod or Jacob's staff, at the head of which is a ball and socket 
joint, enabling a person to move the compass as he may wish. 

(15). The needle is a magnetized steel bar, very delicately 

(20) 



MANUAL OF PLANE SURVEYING. 



21 



balanced upon the point of the center pin, and it is a little shorter 
than the diameter of the compass circle. The delicacy of the 
needle is determined by the number of horizontal vibrations it 
will make before coming to rest after being disturbed. Needles 
5 or 5 J inches in length are generally preferred by surveyors to 
longer or shorter ones. 

(16). Horizontal angles are measured with the compass by 
turning the sights from one line of the angle to the other and 
noting the number of degrees passed over by the end of the needle. 
The sights are sometimes arranged for the measurement of verti- 
cal angles also. 

( 17 ). The letters " E " and "W" are reversed on the compass 
face, but it will be plainly seen that the arrangement enables the 
surveyer to take the direction (bearing) of a line more readily 
from the compass face, and reduces the liability to err in the 
reading. 

( 18 ). This may be illustrated by Fig. 1. 




Fig. 1. 



Suppose the sights set in the direction of the points of the 
circle, and that the compass be turned until the north end of the 
needle marks the point midway between the north point (gener- 



22 MANUAL OF PLANE SURVEYING. 

ally marked with afleur de lis, instead of the letter N) and E, or 
to 45°. The sights have moved in the direction of the second 
hand of a watch, and the bearing of the line marked by them is 
N 45° E, which is read directly from the north end of the needle. 

( 19 ). The following is the general rule for all readings : 

Note the letters between which the end of the needle comes, and 
to what number. Then name the letter N or S (as the case may 
be) that is the nearer to the end of the needle from which you are 
reading, next the number of degrees to which the needle points, 
and lastly the letter E or W that is the nearer to the same end of 
the needle. * 

(20). If the preceding reading had been taken from the south 
end of the needle, the bearing would have been S 45° W, the re- 
verse of N 45° E, but equivalent to it, and indicating the bearing 
taken from the opposite end of the line. 

( 21 ). In taking the northerly bearing of a line, sight from the 
south end of the compass plate, and in taking the southerly bear- 
ing sight from the north end. N 45° E is a northerly bearing, and 
S 45° W is its corresponding southerly bearing. 

(22). In running lines east, the E point of the compass circle 
should be turned toward the north, and in running west, the W 
point should be turned north. Northerly bearings should be read 
from the north end of the needle and southerly bearings from the 
south end. 

(23). In measuring angles observe the following rules: 

1. When both readings are in the same quadrant, as between 
N and E, N and W, S and E, or S and W, the angle is equal to 
the difference between the two readings. Thus, the angle between 
N 56° E and N. 43° E is equal to 13°. 

2. When one reading is in each of either the two north quad- 
rants or the two south quadrants, the included angle is equal to 
the sum of the two readings. Thus, the included angle of N 35° 
E and N 23° W is equal to 35° + 23° = 58°. 

3. When one reading is in each of either the two east quadrants 
or the two west quadrants, the included angle is equal to the sum 
of the readings subtracted from 180°. Thus, the angle included 
between N 50° E and S 37° E is equal to 180°— (50° + 37°) = 93°. 

4. When one reading is in each of two opposite quadrants, the 
angle is equal to the difference of the readings subtracted from 



MANUAL OF PLANE SURVEYING. 23 

180°. Thus, the included angle of N 16° E and S 12° W is equal 
to 180° — (16° — 12°) = 176°. 

( 24 ). The reasons for these rules will be seen in Fig. 2. 




Fig. 2. 

The angle included between the courses A B and A C is equal 
to 59° — 28° = 31°, as both readings are in the same quadrant. 

The angle included between the courses A B and A D is equal 
to 28° + 36° = 64°, because one reading is in each of the two 
Eorth quadrants. 

The angle included between the courses A C and A F is equal 
to 180° — (59° + 33°) = 88°, because one is in each of the two 
<east quadrants. 

The angle included between the courses A D and A F is equal 
to 180° — (36° — 33°) = 177°, because the courses are in opposite 
quadrants. 

(25). The compass box is protected by a glass covering over 
which fits a brass lid. Care should be taken while the brass lid 
is off that no electricity be excited in the glass by the friction of 
the hand or a cloth upon its surface, as it interferes with the work- 



24 MANUAL OF PLANE SURVEYING. 

ing of the needle and may cause a serious error. However, when 
the fluid does exist, it may be removed by breathing on the glass 
or touching it in various places with the moistened linger. 

( 26 ). The action of the needle is also affected by pieces of iron 
or steel brought or kept near it. This materially interferes with 
its use at sea, particularly on iron ships. While surveying, noth- 
ing having a tendency to affect the action of the needle should be 
carried upon the person or allowed near the compass. 

(27). Two kinds of compasses are in use — the sight compass 
and the telescope compass. Each of these may be either a plain 
compass or a vernier compass. The plain compass is not very ex- 
tensively used, as all readings are made from its face alone, and 
can not be depended on for precision. In the plain compass, the 
line of sight lies in the direction of the points of the compass 
circle. The vernier compass differs from the plain compass in 
having its compass circle, to which a " vernier " is attached, mov- 
able, generally through a short arc, about its center, thus enabling 
the surveyor to set the zeros or points of the circle at an angle 
with the line of sight. This angle is read from the vernier. The 
movement of the circle is effected by means of a thumb-screw that 
gives it a slow motion. When the required angle is set off, the 
vernier is clamped to the plate of the compass and the readings 
taken. The vernier enables a surveyor to take a certain class 
of readings ll closer " or with greater precision than is possible 
without it, as it gives him the advantage of a double index : 
yet readings down to a very small angle, say V or SO", are 
hardly ever reliable, owing to the difficulty a surveyor meets in 
setting the needle to exactity. The fault, however, is not in 
the vernier. 

( 28 ). The vernier* consists of an arc divided into a certain 
number of equal parts, and moving within another arc whose di- 
visions are somewhat larger or smaller than its own. The first 
arc (vernier), as stated before, is attached to the compass circle 
and moves with it around a common center ; the second arc is 
called the " limb," and is generally on the brass plate of the com- 
pass upon which the circle moves, so that the outer edge of the 
vernier coincides with the inner edge of the limb. 

-The vernier is so called from its inventor, and on this account the word 
is sometimes written with an initial capital. It was first applied to the 
compass by David Rittenhouse of Philadelphia. 



MANUAL OF PLANE SURVEYING. 25 

(29). Let us now suppose the divisions on the limb to equal 
half-degrees, and that the vernier-arc, corresponding to twenty- 
nine divisions of the limb, is divided into thirty equal parte. 

It is plain, since 30 divisions of the vernier equal 29 divisions 
of the limb, that one division of the limb equals -f-J divisions of 
the vernier, and that one division of the vernier equals §§ di- 
vision of the limb. 

But each division of the limb equals 30 7 or one-half of one de- 
gree ; therefore, one division of the vernier will equal — oa — == ^ > 
which is one minute less than a division of the limb. 

Now, suppose the zero of the vernier to correspond with the 
zero of the limb ; then the points of the compass circle lie in the 
line of sight. If now we turn the vernier until its first division 
from zero coincides with the first division from zero on the limb, 
and on the same side of zero as the division of the vernier, the 
points of the compass will make an angle of V with the line of 
sight; if the second division of each coincide, the angle will be 2 / ; 
if the third, it will be 3 7 , and so on by the same increase, so that 
if we make the twenty-ninth division of each correspond, the an- 
gle will be 29' ; and if we turn still further until the first division 
of the limb coincides with zero of the vernier, the angle will be 
3CK. In the same manner, 3(K acting as a base, the angle may be 
increased to 1°, and so on, 

( 30 ). Sometimes verniers read lower than V\ but they are not 
of much practical use on magnetic instruments. They may also 
differ in construction from the kind described above, but they all 
work on the same principle. The plate on page 26 represents a 
vernier compass; the vernier may be observed on the compass 
plate in front of the box. 

( 31 ). As a general thing, when a compass needs repairing^ 
either from wear or on account of some mishap, it is best to for- 
ward it to some maker of mathematical instruments. This/how- 
ever, may not always be convenient or practicable, and it may be 
well enough to give some directions, which may be of service in 
case of an emergency : 

1. To Re-magnetize the Needle. — "When the needle works lazily. 
on account of losing a portion of its magnetism, it may be re- 
magnetized with a common bar or horse-shoe magnet by passing 
the south pole of the magnet along the north end of the needle 



26 



MANUAL OF PLANE SURVEYING. 



from the center to the extremity and bringing the magnet back to 
the starting point in a circle of five or six inches radius. The 
^outh end of the needle should be treated in the same manner, 
except that the north pole of the magnet should be used on this 
end. From twenty to thirty passes will give it an ample charge. 
2. To Sharpen the Center-pin, — Sometimes the needle moves slug- 
ishly when the center-pin upon which it turns becomes dull. 




When this is the case, take out the plate in which the center-pin 
is set and then unscrew the pin. It may then be sharpened on a 
very fine stone and finished on a piece of smooth leather. Care 
must be taken to grind equally from every side of it. 



MANUAL OF PLANE SURVEYING. 27 

3. To Replace a Spirit-level. — Remove the brass tube from the 
plate and take off the caps at the ends of it. Then with some 
pointed instrument, as an awl or a penknife, scrape out the plas- 
ter or other substance that holds the vial in place, and next force 
out the old. vial by pressing on one end of it. Now slide the new 
vial into place, keeping the proper side up, and if it is too small 
for the tube, wedge it up with pieces of wood or paper. Notice 
carefully its position with regard to the opening in the tube, and 
when it is set in its proper place press some beeswax, boiled plas- 
ter, or putty of the proper consistency, around the ends of it, so as 
to fasten it firmly to the sides of the tube ; then put on the brass 
caps and replace the tube on the compass plate. To re-adjust the 
level, press on the compass plate until the bubble stands in the 
center of the opening in the tube; then turn the compass one-half 
round, and if it remains there, the level is properly placed, but if 
it runs toward the end of the vial, and it probably will, the end 
toward which it settles is too high, and should be lowered or the 
other end raised, whichever is necessary in order to keep the tube 
parallel with the compass plate. After this, give the. compass 
another half-turn and repeat the process given above until the 
bubble will remain in the middle of the opening in the tube in 
«rery horizontal position of the plate. 

4. To Adjust a New Sight. — Fit it to its place on the plate, and 
notice how the slit lines with that of the old one on the opposite 
end. If it inclines to one side, remove it and file off its base on 
the opposite side where it rests on the plate. Then try it again, and 
keep up the operation until the two slits coincide throughout their 
whole length. If both sights need adjusting, hang a plumb, using 
a fine thread or hair, and regulate both sights by it. The com- 
pass should be perfectly level whenever an observation of the 
thread is taken, and the sights will be properly adjusted when- 
ever they correspond with the plumb-line. 

5. To Straighten the Center-pin. — Remove it with its base from 
the rest of the compass and bend it with a pair of pincers or 
wrench made for the purpose, always grasping it about an eighth 
of an inch below its point. 

6. To Straighten the Needle. — It sometimes happens that the nee- 
dle of the compass does not " cut " opposite degrees on the circle, 
as, for instance, when its north point is placed at its south point 
inclines either to the right or left of the opposite 0; when this is 



28 MANUAL OF PLANE SURVEYING. 

the case a portion of the error may be corrected by bending the 
needle with the fingers and the rest by bending the center-pin. 

7. To put in a new Glass. — First take off the brass ring that con- 
tains it (bezzle ring) and remove the pntty. Then take out the 
old glass and put in the new by reversing the process. If the new 
glass is so large that it will not go in readily, hold the edge on a • 
grindstone and grind it down. The manner in which it should be 
ground may generally be seen by noticing the glass just taken out* 

8. The motion of the ball at the head of the Jacob's staff may 
be regulated by a screw-cap that fits down upon it. It should be 
kept reasonably tight in order that the compass may not be too 
easily jarred out of level. If it works loosely, screw the cap down 
tighter. After long usage the ball may not fit the cavity well, 
and in this case it may be taken out and a small piece of sheet 
brass placed under it, or even a piece of paper will answer for a 
short time. 

( 32 ). In carrying the compass it need only be lifted from the 
staff and put under the left arm so that one of the sights may pro- 
ject up behind the shoulder, and the staff makes a good walking 
stick ; but in transportation over the country the sights should be 
taken off and all packed snugly in a box or something else that 
will answer the purpose. A suitable box is usually furnished 
with the compass by the manufacturer. 

( 33). All compasses are provided with a lever or spring with 
which to raise the needle from the center-pin when the compass 
is not in use, and this should not be neglected. 

( 34 ). The compass when not in use should be placed in a hor- 
izontal position and the needle allowed to assume its natural di- 
rection. If this precaution is taken, the needle will better retain 
its polarity. 

(35). The telescope compass is gradually growing into favor 
with surveyors and seems to be taking the place of the sight com- 
pass in many localities. The telescope enables the surveyor to 
set a flag at longer ranges, at greater elevations and depressions, 
and discern it more easily among trees and bushes. It is not quite 
so convenient to handle as the sight compass, however, but this is 
no great disadvantage. Either may be used on a tripod, instead 
of a Jacob's staff. For plate of a telescope compass see frontis- 
piece. This engraving represents the very fine instrument manu- 
factured by T. F. Randolph, Cincinnati. 



MANUAL OF PLANE SURVEYING. 29 

Questions on Chapter II. 

1. Describe the compass. 

2. How are the degrees numbered on the circumference of the 

compass circle ? 

3. How are the sights arranged ? 

4. Describe the magnetic needle. How is the delicacy of a 

needle determined ? 

5. Explain the method of measuring horizontal angles. 

6. How are vertical angles sometimes approximately meas- 

ured ? 

7. Why are the letters E and W reversed on the compass face ? 

8. State the rule for taking the readings from the compass 

circle. 

9. The north end of the needle points 20° to the right of N. 

What is the bearing of the line of sight? What is its 
reverse bearing? 

10. What is a northerly bearing ? A southerly bearing ? 

11. In running lines east what letter on the compass face should 

be turned north? In running west, what one? Why? 

12. Give the four rules for measuring angles. 

13. What is the included angle in each of the following cases : 

N40°EandN62°E? S 15° Wand S 39° E? N29°W 
and S 43° W? 

14. How is electricity excited in the glass of the compass ? How 

may it be removed ? 

15. How T do iron and steel affect the action of the needle? 

16. What two kinds of compasses are in use ? 

17. How many kinds of sight compasses are there ? 

18. What is the difference between a plain compass and a ver- 

nier compass? 

19. Describe the "vernier." Of what advantage is it ? 

20. How is the needle re-magnetized ? 

21. Explain the manner in which the center-pin is sharpened. 

22. How is a spirit level replaced ? 

23. In what way is a new sight adjusted? 

24. How do you determine, when the two extremities of the 

needle do not " cut " opposite degrees, whether it is the 
needle or the center-pin that is bent ? Answer — Turn the 
compass and notice the amount of the error in several 



30 MANUAL OF PLANE SURVEYING. 

positions. If it decreases in certain places and increases 
in others, it is the center-pin. If it remains about the 
same in every position, it is probable that the needle 
alone is bent. 

25. Why should the needle be raised against the glass when 

the compass is not in use ? . 

26. What is the proper position for the compass during the in- 

terval between surveys ? 

27. What are the advantages of the telescope compass over the 

sight compass? 



CHAPTER III. 

THE VARIATION OF THE MAGNETIC NEEDLE. 

( 36 ). The meridian of any point on the earth's surface is a due 
north and south line connecting the point with the poles of the 
earth. 

( 37 ). This is called the true meridian of the place, in order to 
distinguish it from the magnetic meridian r which will be considered 
further on. 

( 38 ). Various methods are employed in determining the true 
meridian, but only two of the most simple and satisfactory will 
be described : 

1. By a Shadow Cast by a Perpendicular Object. — Erect a perpen- 
dicular staff on a level surface, so that its shadow will remain on 




the surface from about 8 o'clock, A. M., till 4 o'clock, P. M., as in- 
dicated in the horizontal projection, Fig. 3, in which S N repre- 
sents the staff. 

(31) 



32 MANUAL OF PLANE SURVEYING. 

Three or four hours before noon ? with a radius, P S, shorter 
than the length of the shadow, and from the point S as a center, 
describe an arc through the point P and produce it beyond, opposite 
P. Then mark the point P, where the shadow last touches the arc, 
and in the afternoon the other point P where it first touches it 
again, and connect these two points with a line PP. Bisect this 
line at O, and the line SO, produced in either or both directions, 
will represent the true meridian of the place. It is not exactly 
correct, except at certain times during the year (at the solstices), 
but it is always sufficiently accurate for ordinary purposes. 

2. By the Polar Star. — The Polar star (Polaris) is situated about 
1J degrees from the north pole of the heavens, and appears to re- 
volve around the pole once in 23 hr. 56 min. If, now, we suppose 
a vertical plane to pass through the north pole and the eye of the 
observer, then twice during the time of revolution Polaris will be 
in this plane, and consequently in the meridian of the observer 
(once when above the pole and again when below it). These are 
called its upper and lower culminations, respectively. 

(a). On the opposite side of the pole from Polaris is a star 
known as Alioth, or more commonly asEpsilon, of the constellation 
of the Great Bear or "Dipper." This is the first star in thefiandle 
of the Dipper, and is situated next to the four that form the quad- 
rilateral, the outside two of which, Dubhe and Merak, are called 
" the pointers," because they indicate the position of Polaris. 

( b ). Since these stars, Polaris and Alioth, are almost exactly 
on opposite sides of the pole, it is evident that they will both be 
on the meridian of the observer when one of them is above the 
other, and it is more convenient to make the observation during 
the upper culmination of Polaris. 

(c). Suspend a plumb-line from some elevated projection, as 
the limb of a tree or a strip nailed to the side of a building or 
high post, and at a point south, not so distant that Polaris will 
rise above the point of suspension of the plumb, arrange a short 
board horizontally east and west, a little below the level of the 
eye. On this board place some kind of a contrivance containing 
an opening across which a thread may be stretched so that it will 
be parallel to the plumb-line when the instrument is in use, and 
slide the instrument along the board until the thread ranges with 



MANUAL OF PLANE SURVEYING. 



33 



the plumb-line and Polaris. Continue to move the instrument 
west as Polaris moves to its point of superior culmination, and 
watch also the approach of Alioth to the meridian, As soon as 
the plumb-line falls on both stars, fasten the instrument to the 



)£ POLARIS. 



NOBm 



PILE. 



>£ ALOTH. 



DUB HE. 



. y^ mermc 



Fig. 4. 

board, and you have two points in the true meridian. The line 
through these may be produced at pleasure and permanently 
marked. Fig. 4 represents the plumb-line covering both stars. 

(d). Still greater accuracy may be reached by following Polaris 
for twenty-two minutes after the plumb-line falls on it, and then 
marking the line. 

(e). When the upper culmination occurs during the day, the 
lower culmination must be used, but Alioth is then very high. 

( 39 ). The following table shows the time of the upper culmin- 
ation of Polaris for each tenth day. The time for intermediate 
days may be approximated by interpolation. The time is given 
to the nearest minute : 
3 



34 



MANUAL OF PLANE SURVEYING. 



Month. 


1st Day. 


11th Day. 


21st Day. 


January . . . 


h. m. 

6 : 30 p. m. 


h. m. 

5 : 50 p.m. 


h. m. 

4 : 50 p. M. 


February . . 


4:27 " 


3:47 " 


2:48 " 


March . . . . 


2 : 32 " 


1 : 53 " 


12 : 50 " 


April 


12:26 " 


11 : 47 a.m. 


10: 49 a.m. 


May 


10 : 29 a.m. 


9 : 49 " 


8:50 " 


June .... 


8:27 " 


7 : 48 " 


6 : 49 " 


July 


6 : 29 " 


5:50 " 


4:51 " 


August . . . 


4 : 28 " 


3:48 " 


2:49 " 


September . 


2:26 " 


1 : 47 " 


12:48 " 


October . . . 


12 : 29 " 


11 : 49 p.m. 


10: 50 p.m. 


November. . 


10 : 27 P. M. 


9 : 47 " 


8 : 48 " 


December . . 


8:28 " 


7:49 " 


6 : 50 " 



(40). If a magnetic compass be placed on the true meridian 
thus established, the needle of the compass, pointing toward the north 
magnetic pole, instead of toward the north pole of the earth, marks a line 
called the magnetic meridian, which coincides with the true merid- 
ian in comparatively few places on the earth. The angle formed 
by the difference in direction of these two lines is called the varia- 
tion or declination of the needle. 

( 41 ). If the north point (south pole) of the needle point to the 
east of the true meridian, it is called east variation, and if it point 
to the west, it is called west variation. The amount of variation is 
determined by the size of the angle. 

( 42 ). In the United States there is a line extending southeast 
through the eastern part of Michigan, western part of Lake Erie, 
eastern Ohio, central West Virginia, Virginia, and North Carolina, 
reaching the Atlantic ocean near Wilmington, that is called the 
agonic line or line of no variation, because at all points thereon the 
magnetic meridian and true meridian coincide. Places east of 
this line have west variation, and those west of it have east varia- 
tion. 



MANUAL OF PLANE SURVEYING. 35 

(43 ). Isogonic lines or lines of equal variation run through places 
having the same variation. For instance, the line of three degrees 
west variation passes through Chesapeake Bay, Maryland, central 
Pennsylvania, and western New York, and the line of three de- 
grees east variation passes through western South Carolina, eastern 
Georgia, Tennessee, Kentucky and Indiana, and western North 
Carolina, Ohio, and Michigan. The variation increases in both 
directions from the agonic line, reaching about 18 degrees west 
variation in eastern Maine and 22 degrees east variation in the 
northern part of Washington Territory. 

( 44). The isogonic lines converge toward the north magnetic 
pole, situated at present in longitude about 96° west from Green- 
wich, and latitude about 70° north. This pole has been gradually 
moving westward for several years, and will perhaps continue to 
do so for several years to come. This causes the agonic line and 
all the isogonic lines to move in the same direction, so that west 
variation is constantly increasing and east variation constantly 
decreasing. 

(45). The change that takes place in the variation of the 
needle at any place from this cause is called its secular change. 
This varies in different localities, and is generally greater in the 
northern part of the United States than in the southern part. It 
is determined by comparing the variation at the time of any ob- 
servation with that of a preceding or succeeding one. 

( 46 ). But this is not the only change to which the needle is 
subject. Its action is modified by other influences. The north 
end of the needle moves westward from about 6 o'clock A. M., 
until about 2 o'clock p. M., and then gradually returns to the 
starting point. This is called its diurnal change, and it sometimes 
amounts to 10 or 12 minutes of a degree. 

This change is about twice as great in summer as in winter, 
hence an annual change must be taken into consideration. 

( 47 ). The following table, taken from the Keport of the United 
States Coast Survey, illustrates these changes. The mean magnetic 
meridian is the average position of the needle for the day : 



36 



MANUAL OF PLANE SURVEYING. 



Houe. 


G 
ft 


S 
g 


8 

5 


'I 




6 A. M. 

7 " " 

8 " " 

9 " " 
10 " " 


i 

3 

4 
4 

1 


4 
5 
5 

4 
1 
2 
4 
6 
5 
4 
o 

2 

1 


2 

3 
3 
2 




1 
1 
2 
2 
1 


3 

35 


11 " " 

12 M. 

1 P. M. 
o « ft 

3 " " 

4 <• << 

5 " " 

6 " " 


1 
4 
5 
5 
4 
3 
2 
1 


2 

3 
4 
3 
2 
1 
1 




2 
3 
3 
2 
1 
1 



o 

3 © 

go p, 

PS" 

• » 

c 
3 

CD 



(48). Probably as good a time as any to get the variation of 
the needle is between 5 and 7 o'clock in the evening, as it then 
pretty nearly indicates the mean magnetic meridian. The diur- 
nal and annual changes are usually disregarded in practical work 
with the compass. 

(49). The needle is also subject to disturbances that do not 
appear conformable to any known law. These generally take 
place during a thunder storm, aurora, or other electrical phe- 
nomenon, and sometimes cause a surveyor no small amount of 
vexation. 

(50). The north end of the needle is also inclined to point 
downward. This is called its "dip" or inclination, and increases 
as we go northward. It is overcome by making the north end of 
the needle the lighter, and in order to render the needle adjusta- 
ble, a small counterpoise is arranged in a groove on its south end. 

(51). A great many magnetic phenomena are at present very 
imperfectly understood, and many of the conclusions based upon 



MANUAL OF PLANE SURVEYING. 37 

partially accepted theories are liable to be overthrown at almost 
any time. A discussion of these belongs to Philosophy. 

Questions on Chapter III. 

1. What is meant by the meridian of a place? 

2. What is the difference between the true meridian and the 

magnetic meridian ? 

3. Explain each of the methods given for establishing a true 

meridian. 

4. What is meant by the culmination of Polaris? 

5. To what does the north end of the needle point ? 

6. What do you understand by " the variation of the needle? " 

7. What is east variation? West variation ? 

8. What determines the amount of variation? 

9. Through what part of the United States does the agonic 

line run? 

10. What are isogonic lines? 

11. What is the situation of the north magnetic pole? 

12. In what direction is the agonic line and all the isogonic 

lines moving at present? 

13. What effect does this movement have on east variation? 

West variation ? 

14. What other influences act upon the needle? 

15. What is the difference between the diurnal change and the 

annual change? 

16. What is meant by " dip ? " 

17. How is it overcome ? 



CHAPTER IV. 



EFFECT OF CHANGE OF VARIATION- ON OLD LINES AND METH- 
ODS OF CORRECTING BEARINGS. 




Fig. 5. 



-The terms variation and declination have become synonymous in mean- • 
ing, although it would be better, etymologically speaking, to use the word 
variation to denote the change of declination. We would then say variation 
of declination, instead of change of variation as at present. 



(38) 



MANUAL OP PLANE SURVEYING. 



39 



( 52 ). As the variation of the needle is constantly changing, so 
the bearing of a line surveyed at any time is also subject to con- 
stant change. This renders it necessary in the re-survey of any 
line to take into consideration the amount of change that has taken 
place since the previous survey ; otherwise the location of the line 
will be changed. 

To illustrate : Let the line A B, Fig. 5, represent the line of 
direction of the magnetic needle (magnetic meridian) at a certain 
time, and the line C D which makes an angle of 22° with the mag- 
netic meridian, and whose course is N 22° E, represent a line sur- 
veyed while the needle marks this meridian. 

Since the north end of the needle is continually moving toward 
the west, it is evident that the angle between A and C, formed by 
the crossing of the two lines, will be constantly growing larger, 
and that the bearing of the line C D will consequently increase 




Fig. 6. 



40 



MANUAL OF PLANE SURVEYING. 



with each succeeding year, as long as the westward movement of 
the needle continues. 

( 53). Let us now suppose that several years after this survey 
is made, the needle assumes the position of the dotted line E B 7 
Fig. 6, making an angle with A B equal to 5°. The line C D will 
now make an angle with the magnetic meridian equal to (22° -f- 
5°) = 27°. It will be seen that the bearing of the line has in- 
creased 5°, the change of variation of the magnetic needle. 

(54). If now the surveyor, through ignorance or carelessness, 
should neglect to take this change into consideration, and run the 
line at the old bearing (22°) marked in the description of the line, 
he would change its position to that of the dotted line D F, and 
it is easy to see what an error this would cause in the survey of 
a tract of land. 

( 55 ). A survey of this kind, however, does not affect the form 
or area of a piece of land, but simply changes its boundaries and 
seems partially to revolve the tract around the corner from which 
the surveyor starts, as a center. For instance, suppose a mistake 



A...J. 




Fig. 7. 



of this kind to be made in the survey of the tract ABCD, Fig. 
7; its position would be changed to that of the dotted quadri- 
lateral. In this survey the corner D was made the starting point. 
( 56 ). In order to determine the present bearing of a line, it is 
an advantage to know three things: 1. The bearing of the line 



MANUAL OF PLANE SURVEYING. 41 

at the time of a previous survey. 2. The number of years that 
have elapsed since that survey was made. 3. The annual amount 
of secular change of variation. 

1 and 2. Where field-notes of the former survey have been pre- 
served, they generally and should always, give the necessary in- 
formation in regard to the bearing of the line and date of the sur- 
vey ; but where no record of the survey can be found, the bearing 
may sometimes be approximately determined from old deeds or 
descriptions of the land, and these frequently assist the surveyor 
in arriving at a conclusion with regard to the time the survey 
was made. Persons living in the neighborhood may also know 
something of the previous survey. 

3. In order to determine the annual change of variation, vari- 
ous methods may be employed. 

(1). The bearing of a line at the present time may be compared 
with that recorded by a competent surveyor some time previous. 

(2). A true meridian may be established by the second method 
explained in the preceding chapter, and the variation of the needle 
found from this by observing the angle the magnetic meridian 
makes with the true meridian. This result may then be com- 
pared with the variation recorded after a previous observation. 

In both of these cases the amount of change in minutes will 
equal the quotient obtained by dividing the difference of varia- 
tion in minutes between the two observations by the number of 
years that have elapsed between them. For instance, if the varia- 
tion of the needle at a certain time is found to be 5°, and in twenty 
years after an observation shows that it has decreased to 4°, the 
annual change will equal (5° — 4°) = 1° = 60', which divided by 
20 = 3'. From this the variation for any subsequent time may 
be found by multiplying the annual change, 3', by the number of 
years, and adding or subtracting the product, as will be explained 
further on. 

(3). When the annual change at several important points is 
known, the change at intermediate points throughout the country* 
may be determined by interpolation, but this method is not relia- 
ble in all cases. 

( 57 ). The following table is taken, somewhat modified, from 
the report of the JJ. S. Coast Survey of a few years ago, and con- 



42 



MANUAL OF PLANE SURVEYING. 



tains the variation of the needle at various important stations 
throughout the United States for the year 1881 : 



Bangor, Me., 16° SV W. 

Wolfsboro, N. H., 12° 15' W. 

Middlebury, Vt., 11° 35' W. 

Marblehead, Mass., 1?° 54' W. 

Schenectady, N. Y., 9-58' W. 

Chambersburg, Pa., 4° 33' W. 

New Haven, Conn., 9° 00' W. 

Providence, R. L, 11° 27' W. 

Wilmington, N, C, 1° 04' W. 

Jackson, Miss., * . . 6° 33' E. 

Washington, B.C., 3° 17' W. 

Camden, N. J., 6° 50' W. 

Columbus, O., 0° Z& E. 

Savannah, Ga., 1° bOf E. 

New Orleans, La., 6° 33' E. 

Quincy, Wis., :\ . . . 6° 33' E. 

Decatur, Ind., 1° 50" E. 

Grand Rapids, Mich., 1° 5C E. 

Havana, Cuba, 4° 46' E. 

Carlinville, 111., G° 33' E. 

Aiken, S. C, 1° off E. 

Monterey, Cal., 16° 06' E. 

Sitka, Alaska, 28° 27' E. 



This list is not full enough to be of much practical use, but I 
think best not to extend it any further, as the surveyor can easily 
find the variation at any point himself. 

( 58 ). The next table, also taken from the Keport of the Super- 
intendent of the Coast Survey, gives the annual amount of secular 
rkange in certain localities in the United States. 
* Maine, Long Island, Delaware, Maryland, 
Virginia, South Florida, South Alabama 

and Mississippi, and New Jersey 3' 

East Maine, West Tennessee, Missouri, and 
West Louisiana 2 / 



MANUAL OF PLANE SURVEYING. 43 

Ohio, East Tennessee, East Louisiana, and 

East Massachusetts 2' 15" to 2' 45" 

New Hampshire, West Massachusetts, Khode 

Island, Connecticut, West Virginia, North 

Carolina, South Carolina; Georgia, *and 

North Florida 3' 15" to 3' 45" 

North and West New York 4 / 30" 

Vermont 5' 30" 

West Louisiana Y 

In all these places the change marked to the right indicates an 
increase of westerly or decrease of easterly variation. 

( 59 ). These tables will enable us to approximate to the vari- 
ation of the needle at points included in both tables, for perhaps 
several years. For example, the variation of the needle at Cam- 
den, N. J., this year is 6° 50 7 W; in 1885 it will be about 6°50 / + 
(4 X 3 / ) = 7° 02'. If the variation is easterly, as in the western 
part of the United States; the amount of change is subtractive, 
instead of additive. Take this for example: What will be the 
variation of the needle at New Orleans in 1884? 

1884—1881=3 years. 

Variation in 1881 = 6° 33'. 

Annual change about 2' 30". 

Amount of change = (3X2 ; 30")=7' 30". 

Variation in 1884 = 6° 33'— 7' 30° = 6° 25' 30". 

(60). In order to find what the variation of the needle was a 
few years ago, we have only to reverse the rule given ; that is, add 
where the variation is easterly, and subtract where it is westerly. 

The following examples may be used for practice : 

( 1 ). The variation of the needle at a certain place in 1880 was 
N 5° 32' E, and the annual change 2 / 20" ; what will be the vari- 
ation 1889? 

( 2 ). In 1878 the variation at a certain place was 4° 15 / W, 
annual change 3' 30"; what will be the variation in 1886? < 

(3). In 1880 the variation at A was 2° 36' W; what was the 
variation in 1874, supposing the annual change to be 4'? 

( 4 ). The annual change at B is 3 / 45" and the variation of the 
needle in- 1879, 3° 47' E; what was the variation at this place in 
1870? / 



44 



MANUAL OF PLANE SURVEYING. 



( 01 ). It would frequently be better if the bearings of lines 
were taken from the true meridian as a basis, as no change would 
then take place in the bearings. 

This may be easily explained. Suppose the line A B to repre- 



A C 




Fig. 8. 



sent a section of a true meridian of the earth, and the line C D to 
represent one of the boundaries of a piece of land. Both of these 
lines are fixed, and consequently the angle between them can 
never change. 

( 62 ). If, however, the line A B represents a section of the mag- 
netic meridian, it is constantly changing, and this of course affects 
the angle. 

( 63 ). In order to determine the true bearing of a line from its 
magnetic bearing, two cases must be considered : 

1. When the variation is west. Rule. — Add the variation to the 
bearing for northwest and southeast courses, and take their differ- 
ence for northeast and southwest courses. 

In Fig. 9 let A B represent a due north and south line and C B 
represent the direction of the needle. D B and E B are two lines 
whose bearings are to be changed from the magnetic bearing to 
the true bearing. Suppose the variation of the needle to be 10° 
W; then the true bearing of the line E B will be N (25° + 10°) 



MANUAL OF PLANK SURVEYING. 



45 



W = N 35° W, and the true bearing of the line B D will be N 
(35° — 10°) E = N-25°E. 




Fig. 9. 



2. When the variation is east. Kule. — Reverse the preceding op- 
eration ; that is, take the difference between the bearing and vari- 
ation for northwest and southeast courses, and their sum for north- 
east and southwest courses. 

The line A B, Fig. 10, represents a section of the true meridian, 
and the line B C a section of the magnetic meridian. As before, 
the bearings of the lines D B and E B are to be changed to the 
true bearings from the magnetic bearings. If the variation of the 
needle be 10° E, the true bearing of the line B D will be N (25° 
+ 10°) E = N 35° E, and the true bearing of the line B E will be 
N (35° — 10°) W = N 25° W. 



46 



MANUAL OF PLANE SURVEYING. 




(64). The variation in the following examples is 5°; what ia 
the true bearing of each course ? 

1. N 45° W. 

2. N 32° W. 

3. S 16° W. 

4. N 17° E. 

5. S 25° E. 

6. N 25° E. 

(65). If in any case the sum of the variation and bearing 
amounts to more than 90°, the supplement* of the sum must be 
taken, and the first letter changed to its opposite. 

Take, for instance, the bearing N 88° W when the variation is 
6° W. This gives us the following result : 88° + 6° = 94°, and 
taking 94° from 180° to get the supplement, we have (180° — 94°) 

-The supplement of an arc is what remains after subtracting the arc from 
180°. 



MANUAL OF PLANE SURVEYING. 47 

== 86°. After changing the first letter of the magnetic bearing^ 
we see that the true bearing of the line is S 86° W. 

( 06 ). In changing from the true to the magnetic bearing, it is 
necessary only to reverse the foregoing rules. 

Questions on Chapter IV. 

1. What effect does the change of variation of the magnetic 

needle have upon the bearings of lines? 

2. Why must this change be taken into consideration in a re- 

survey ? 

3. Why would it not affect the figure or contents of a tract of 

land if this change were not regarded ? 

4. What three things are necessary in order to determine the 

present bearing of a line ? 

5. How is its bearing at the time of a former survey found ? 

The annual change ? 

6. Two observations thirty years apart show a change of 1° 15 A 

in the variation of the needle. What is the annual change? 

7. What is meant by east variation ?. West variation. 

8. How do you find the true bearing of a line from the mag- 

netic bearing ? The magnetic from the true ? 



CHAPTER V. 

METHOD OF RUNNING LINES, ETC. 

( 67 ). The first thing a surveyor does in making a survey is to 
find a corner from which to start, and this is frequently a matter 
of no little trouble, but its discussion belongs further on in our 
work. Let us suppose for the present that the monument which 
marks the corner (generally a stone or a stake) is still standing, or 
that the witnesses to it are yet to be found. These witnesses are 
generally trees, although any other immovable object will answer 
the purpose equally well. When the corner is put down the course 
and distance from it to one or more of these objects is carefully 
noted in the record (field notes) of the survey, so that, no matter 
if the monument that marks the corner has disappeared, its posi- 
tion can easily be determined as long as any remains of the wit- 
nesses exist. The witnesses, if trees, are called witness-trees, 
and are marked about eighteen inches from the ground on the side 
facing the corner. The mark consists of a blaze about six or eight 
inches long, which generally has a cross notch cut in it. 

( 68 ). If the monument at the corner remains, its center should 
be taken as the point from which to run the line ; but if it can not 
be found, its position must be determined from the witnesses. It 
is probable that the approximate location of the corner is known, 
so that the surveyor or some of his attendants know where to look 
for it. Suppose the surveyor examines his record and finds that 
when the corner was set, a beech tree 18 inches in diameter was 
taken as a witness to it. This tree stood in a direction N 43° E 
from the corner, and was distant 78 links. He looks in that direc- 
tion and sees a tree which he believes to be the one, and after an 
examination he finds the surveyor's mark and is convinced of its 
identity. All he has to do now is to set the compass in front of the 

(48) 



MANUAL OF PLANE SURVEYING. 49 

mark on the tree and turn it until the needle indicates the reverse 
bearing, S 43° W; a staff is then set in the line of sight somewhat 
further from the tree than the distance given for the corner, and a 
measurement of 78 links made from the notch on the tree in the 
direction of the staff; the termination of this measurement is the 
corner. 

(69). On prairies, where no trees are available for witnesses, 
mounds of earth thrown up around a stake are usually made to 
answer the purpose, and in clearings it is sometimes necessary to 
make the same kind of witnesses, or supply their places with stones 
set at suitable distances from the corner. A good monument 
should always be put down at the corner, and one not easily 
moved or destroyed does not particularly need any witnesses to it. 

(70). The method of describing a witness-tree employed by 
surveyors in making up their field-notes, is much abbreviated from 
the method used above. The above description would read as fol- 
lows : Be 18 N 43 E 78. The first number signifies the diameter, 
the second the bearing, and the third the distance in links. The 
bearing and distance of the tree from the corner are called its 
u course and distance." Likewise the course and distance of any 
line are its bearing and length. 

( 71 ). After the corner has been found, the next thing is to run 
a line from this corner to the next. In surveying, a line always 
connects two corners, and in plane surveying it represents* the 
shortest distance between them. 

( 72 ). In order to run this line, its course and distance must be 
at least approximately known, and should be exactly known. 

( 73 ). Let us suppose that the true bearing of the line is N 3° E, 
and its length 6 chains and 50 links. Since the true bearing is 
given it has remained unchanged (Art. 61), and the surveyor has 
only to turn the vernier of the compass until he sets off the varia- 
tion of the needle, and then set the north end of the needle at N 
3° E. The sights of the compass are then set on the line, and a 
measurement of 6 chains and 50 links on the line of sight will 
bring him to the next corner. 

(74). In setting off the variation of the needle on the vernier, 
two cases arise :'(1) when the variation is east, and (2) when the 
variation is west. 

1. To Set Off East Variation. Eule. — Turn the sights to the left 
4 



50 MANUAL OF PLANE SURVEYING. 

until the angle between the line of sight and the points of the 
compass circle equals the variation of the needle. 

2. To Set off West Variation. Kule. — Turn the sights of the com- 
pass to the right until the angle between the line of sight and the 
points of the circle equals the variation. 

These rules obtain whether the movement of the needle is to- 
ward the west or east. 

We are now prepared to run lines whose true bearings are 
known. 

The course and distance of a line are N 43° E 16 chains and 50 links, 

(75). After finding the first corner on this line, the surveyor 
sets off the variation of the needle on the vernier, and then sets 
the compass at the corner, levels it, and turns the sights until the 
needle indicates the bearing of the line. He then starts the flag- 
man in the direction of the line of sight to a suitable distance 
from the compass, and then moves him either to the right or left 
until the sights strike the flag-staff. The staff is left in this posi- 
tion until the surveyor comes up and sets the compass where it 
stood, and then the flagman seeks another position further along 
the line. 

( 76 ). The chain-men commence at the center of the monument 
that marks the corner as soon as the surveyor has set the flag on 
the line. The one that takes the lead is called the " leader," and 
the one at the rear end of the chain is called the " follower." At 
setting out, the leader takes a certain number of iron or steel pins 
(usually ten) and puts one down at the end of the chain. The 
chain is then carried forward another length and its rear end held 
against the pin just put down while the leader sticks another into 
the ground. Great care should be taken to keep the chain free of 
" kinks," taut, and as nearly horizontal as possible, no matter how 
uneven the ground may be, and the follower should line each pin 
with the flag-staff or compass, whichever may be set at the point 
to which they are running. 

(77) When ten pins are used the "marker" drives a stake* 
where the last pin stood ; the leader takes the ten pins from the 
follower, and the measurement proceeds just the same as from the 
corner. If eleven are used the follower retains one in starting out 
from the corner, and as soon as the leader puts down the last of 

-The points on the line marked by the stakes are called " outs," because 
the leader runs out of marking pins at these places. 



MANUAL OF PLANE SURVEYING. 51 

his ten lie receives ten from the follower, and the new measure- 
ment begins at the last pin put down, instead of at the stake, as 
before. As we said before, the chain usually employed in survey- 
ing is only two rods, fifty links, or thirty-three feet in length, and 
since a stake is put down at every ten lengths, it leaves the stakes 
twenty rods, or five chains apart. The first stake from the corner 
has one notch cut in it, the second two, the third three, and so on. 
The surveyor should sight back while the marker is setting the 
stake and see that it is put exactly on the line. 

( 78 ). The relative places on the line of the persons engaged in 
the survey are as follows : (1) flagman, (2) surveyor, (3) chain 
carriers and marker. 

( 79). The measurement is continued in the direction N 43° E 
until the length of the line, 16 chains and 50 links, has been meas- 
ured, and if the line thus run strikes the corner, it coincides with 
the true line, but if it does not, the stakes must be moved, either 
to one side or the other, until it is made to coincide. 

( 80). Let us suppose that in the case before us the line has 
been found to terminate at a point 8J links to the left of the cor- 
ner. This line is called a " random line," and it is evident that 
it has been diverging from the true line gradually since leaving 
the starting point, and that this divergence is always in propor- 
tion to the distance from this point. 

Dividing now the distance between the terminations of the two 
lines by their length in chains, we have 8£ -*- 16J = J link for 
the increase of divergence for each chain from the point from 
which the lines start, and since the first stake is 5 chains from 
this point, it is 5 X i link = 2J links off the true line. In like 
manner the second stake is 10 X J link = 5 links off the true 
line, and the third one 15 X i link = 7 J links off the true line. 
The stakes are put on the true line by moving them the given dis- 
tances in the proper direction which is always the opposite of the 
direction of the termination of the random line from the corner. 

( 81 ). This may be illustrated by the figure. 



>% 



2 


3 _ V 




n — n 


5 
Fig. 11. 


r/ 2 u 



52 MANUAL OF PLANE SURVEYING. 

Let A C represent the true line, and A B the random line. 
Since the termination of the random line lies to the left of the true 
corner, the stakes must be moved to the right. The numbers on 
the random line indicate the order in which the stakes are num- 
bered, and those on the true line represent the distance that each 
stake must be moved to the right. 

( 82 ). This enables us to deduce the following rule for correcting 
the stakes: Divide the distance between the terminations of the 
two lines in links by their common length in chains, and multi- 
ply the quotient by the number of chains the stake is distant from 
the starting point. The product will equal the number of links 
the stake is to be moved. Then move the stake in a direction op- 
posite to that of the termination of the random line from the 
corner. 

Where the length of the line is a multiple of five chains (the 
distance between stakes), the following rule will be found more 
simple : Divide the distance between the terminations of the two 
lines in links by the number of stakes on the line, and multiply 
the quotient by the number of the stake from the starting point. 
The product indicates the number of links the stake is to be moved 
as before. 

( 83 ). In both of these rules the terminations of the two lines 
and the starting point are considered as the vertices of an isosceles 
triangle, and the distance between the terminations of the two 
lines should be measured on a line that will make the same angle 
with one as with the other. 

(84). In the following examples, give the distance and direc- 
tion each stake should be moved : 

(1). Length of line, 20 chains. Ran to the left of true corner 
40 links. 

(2). Line 40 chains long. Ran to the left 20 links. 

(3). Line 20 chains. Ran to the right 30 links. 

(4). Line 18 chains. Ran to the left 12 links. 

(5). Line 16 chains. Ran to the right 32 links. 

(6). Line 26 chains. Ran to the right 45 J links. 

( 85 ). In making up his field-notes, the surveyor usually writes 
links as hundredths of a chain. For instance, ten chains and 
forty links would be written 10.40. He also sometimes uses the 
word " missed " to denote that the termination of the random line 
lies either to the right or left of the true corner, and accompanies 



MANUAL OF PLANE SURVEYING. 5& 

it with the word " right v or " left," as the case may be. These 
terms are frequently abbreviated in writing by using their initial 
letters instead of the words themselves. Thus, " missed left " may- 
be cut down to M. L. 

( 86 ). Owing to the imperfection of magnetic instruments it is 
common, particularly on long lines, for a surveyor to run either 
to one side or the other of the corner to which he is running, even 
when he knows the bearing of the line. In this case, however, he 
seldom misses the corner any considerable distance — perhaps only 
two or three links — and the bearing he has taken may be assumed 
to be correct, the error being due to the manipulation of the in- 
strument. 

( 87 ). But where the bearing is only approximately known, the 
random line may vary greatly from the true line, and when this 
happens the assumed bearing must be corrected in order that the 
true bearing may be subsequently determined without a resurvey / 
of the line. ^ 

( 88 ). Suppose a line 40 chains long is to be surveyed, and 
from the best evidence the surveyor has, he assumes its bearing to 
be N 2° E. The line is accordingly surveyed, and upon reaching 
the other end the surveyor finds that he has missed the corner 70 
links. It is plain that he assumed a bearing considerably in 
error, and he wishes to correct it. To do this, he may employ 
either of the following rules : 

1. Multiply the length of the line by .01745 and divide the pro- 
duct by the product of the distance between the extremities of the 
two lines by 60. The quotient will be the correction in minutes 
of a degree. In this case we have , 

(.01745 X 40) -*- (70 X 60) = 60 / + = 1° +. 

All distances should be in chains and hundredths of a chain. 

This rule is derived as follows : In Fig. 12 the lines A B and A. 
C represent the true line and random line, respectively, of the 
survey. C B represents the distance between their extremities. 

(1). The line C D is called the sine of the angle B A C, and where 
the angle is small, it does not vary much in length from the chord 
of C B which connects the extremities of the two lines. The line 
A D is called the cosine of the angle BAG, and where the angle 
is small it does not differ materially in length from the line A B. 

(2). The sine of an angle, therefore, is a perpendicular raised 



54 MANUAL OF PLANE SURVEYING. 




from one of its sides to the other, and the cosine of an angle is the 
portion of one of its sides intercepted between the foot of the sine 
and the vertex of the angle itself. 

(3). Particular attention should be paid to the relation of the 
sine and cosine to one another, and they should also be studied in 
their application to the lines of the survey, as explained above. 
The surveyor needs to employ sines and cosines frequently. 

(4). If now we call the length of the radius 1, the length of the 
sine of an angle of o*ie degree will be .01745 which is the number 
we employed in the rule. The sine will increase in the same ratio 
that the radius increases. Therefore, if the radius is 40, the sine 
will be (40 X .01745) = .69800 ; and as this is to the distance be- 
tween the extremities of two lines, so is 60, the number of minutes 
in a degree, to the number of minutes of correction ; hence the 
rule. 

(5). Where the angle is large, the sine may differ considerably 
in length from the chord, but this is a case that will seldom come 
up in practice, except where the surveyor chances to make a mis- 
take, and in this case he would better resurvey the line. 

2. In rectangular surveying, nearly all the lines to be surveyed 
in usual practice in dividing and sub-dividing the section, are J 
mile, J mile, or one mile in length. Now, the sine of an angle of one 
degree for a radius of 20 chains, or J mile, is very nearly 35 links ; 
therefore, for 40 chains, or \ mile, it is (2 X 35) = 70 links ; and 
for 80 chains, or 1 mile it is (4 X 35) = 140 links. 

(1). This enables us to modify the rule already given, as follows : 



MANUAL OF PLANE SURVEYING. 55 

f Multiply the distance between the extremities of the two lines by 
60, and divide the product by the sine of one degree for a radius 
equal to the length of the line surveyed. The quotient will be 
the number of minutes of correction to be made. 

(2). If, for example, in a line J mile long, the surveyor miss 
the corner 70 links, the correction is made as follows : 

e 70 : 70 : : 60 : x = 7Q * 60 . = 60' = 1°. 
/0 

3. If, however, he has its bearing at a previous time, he may 
then find its present bearing by the following rule : Kun a line 
from one extremity of it with the old bearing and distance. Meas- 
ure the distance between the other extremity of this random line 
and the true line. Multiply this distance by 57.3 and divide the 
product by the length of the line surveyed. The quotient will be 
the change of variation expressed in degrees. 

Suppose the length of the line to be 18.25 and the distance be- 
tween the extremities 35 links. Then, 57.3 X -35 = 20.055, and 
20.055 -r- 18.25 = 1.09° = 1° 5' 24".* 

( 89). In determining the correction to be made in the follow- 
ing examples, use whichever one of the preceding rules appears 
most expeditious: 

(1). Length of line, 16 chains. Distance between terminations 
18 links. 

(2). Length of line, 36 chains. Distance between extremities, 
32 links. 

i 3). Length of line, 20 chains. Bearing 7° 32'. Distance be- 
tween extremities, 30 links. 

(4). Length of line, 40 chains. Distance between extremities, 
48 links. Bearing 16°. 

( 90). Let us now see whether this correction to be made is to 
be added to the assumed bearing of the line, or subtracted from it. 

This involves two cases : 

-Suppose the random line and true line to represent two radii of a circle, 
and the line connecting their extremities to be a portion of the circumfer- 
ence of the same circle. We shall then have this proportion : 

Whole circumference : arc : : 360° : angle of lines. 

Whence, in case under consideration, we have 
(36.50 X 3.1416) : 35 : : 360° : 1.09°. 

To generalize, let A represent the starting point of the lines and B and C 
their extremities. The angle formed by them will be expressed by B A C. 

Then 2 A B X 3.1416 : B C : : 360° : B A C. 

-p r\ 

Whence, BAC = -r-= X 57.3, very nearly 



56 



MANUAL OF PLANE SURVEYING. 



lc Iii Northeast and Southwest Courses. Bule. — Add the minutes 
of correction to the assumed bearing of the line (bearing of ran- 
dom line) when the random line lies to the left of the true line, 
and subtract when it lies to the right. The sum or difference will 
be the true bearing of the line. 

To illustrate, let the line A B, Pig. 13, represent a section of the 
true meridian, A C the line to be surveyed, A D a random line 
lying to the left, and A E a random line lying to the right of the 
true line. 




Fro. 13.. 

The fact is apparent that the bearing of the line A C is equal to 
the bearing of the line A D plus the angle D A C; and it is 
equally apparent that the bearing of the same line, A C, equals 
the bearing of the line A E minus the angle C A E. 

2. In Northwest and Southeast Courses. Kule. — Reverse the pre- 
ceding rule; i. e., subtract the minutes of correction from the as- 
sumed bearing (bearing of random line) when the random line lies 
to the left of the true line, and add when it lies to the right. 



MANUAL OF PLANE SURVEYING. 57 

(91). In the following examples the bearings of the random 
lines are given, and the words "right" and "left" signify their 
position with regard to the true line. What is the bearing of the 
true line in each case? 

(1). N 40° E 40.25, left .40. 

(2). S 32° W 16.50, " .25. 

(3). N 71° W 14.00, right .12. 

(4). N 15° 30' E 8.40, " .06. 

(5). S 5° 15' W 20.00, " .13. 

(6). S 6° 17' E 20.03, left .13. 

(7). N 4° 06' W 15.50, " .23. 

(8). N 3°21 / E 40.00, right .10. 

The distances are all given in chains and hundredths of a chain. 

( 92 ). So far, we have based all our bearings on the true merid- 
ian, but in many cases lines are surveyed and their magnetic bear- 
ings, instead of their true bearings, -given. 

(93). When this is the case, and a resurvey is to be made of 
these lines, it is necessary to know what change has taken place 
in the variation of the needle since their bearings were determined, 
because the bearings change with the variation. Having found 
this, their present bearings may be determined by the following 
rule: 

1. To Correct Magnetic Bearings. — In northeast and southwest 
courses, add the change to the given bearing, and the sum will be 
the present bearing. In northwest and southeast courses, sub- 
tract the change from the bearing, and the difference will be the 
present bearing. 

2. In Fig. 14 let A B and A C represent two courses, one bear- 
ing northeast and southwest, and the other northwest and south- 
east. Also, let A D represent the direction of the needle when the 
lines were surveyed, and A E its direction at the present time. 
The change of variation equals the angle E A D, and it will be 
seen that the present bearing of the line A C is equal to the sum 
of D A C and DAE. Likewise, that the present bearing of the 
line A B is equal to the difference between DAB and DAE 

( 94). This rule holds good while the needle moves toward the 
west. If its movement change, the rule will have to be reversed. 



MANUAL OF PLANE SURVEYING. 




Fig. 14. 



( 95 ). Correct the bearing of each of the following courses for 
the present year : 

(1). N 14° E, annual change 2' 30". Survey made in 1856. 

(2). N 7° 34' W, annual change 3'. Survey made in 1862. 

(3). S 1° 13 / E, annual change 2\ Survey made in 1874. 

(4). S 3° 02' W, annual change 2 / 15". Survey made in 1858. 

( 96 ). Sometimes the letters T. M. are placed after a bearing to 
denote that it is based on a true meridian, and M. M, sometimes 
follows a magnetic bearing to show that it is based on the mag- 
netic meridian. These precautions are necessary, and their omis- 
sion is frequently a cause of much perplexity to surveyors. 

After the bearing of the line has been determined, the survey of it is 
entirely similar to the survey of lines whose true bearings are given, 

( 97 ). When the surveyor finishes a random line, he generally 
walks back along it and moves all the stakes to the true line, and 
perhaps marks trees that stand on or near the true line, so that it 
mav be the more easilv found in he future. 



MANUAL OF PLANE SURVEYING. 59 

He then enters its course and distance in his field-notes, or sur- 
veyor's record, and this completes the survey, although he may 
sometimes draw a plot of the survey, particularly if it is of a tract 
of land. 

(98). Flag-men, chain-men, and markers are usually sworn 
before the survey begins. 

( 99 ). The surveyor should be careful about keeping his chain 
of the proper length by testing it frequently with a standard 
measure, and should watch his assistants closely until they un- 
derstand what is required of them. 

It would also be well for every county to have a true meridian 
established, so that the variation of the needle might be found at 
any time, and with but little trouble. 

(100). Back sights should always be taken at intermediate 
stations along the line in order to avoid possibility of deflection. 

Questions on Chapter V. 

1. What is a "witness?" 

2. How are witnesses marked? 

3. How are corners found from witnesses? 

4. What is meant by the "course and distance" of a line? 

5. What must we know before we can run a line? 

6. How do you set off east variation on the compass vernier? 

West? 

7. Of what length is the chain usually employed in surveying? 

8. If the random line lies to the right of the true line, in what 

direction do you move the stakes? 

9. Give the rule for correcting the stakes. * 

10. How should the distance between the extremities of the true 

line and the random line be measured? 

11. How is 5 chains and 12 links usually wi^tten by a surveyor? 

16 chains and 4 links? 

12. Give the first rule for finding the amount of error in as- 

sumed bearing. The second. 

13. What is meant by sinef Cosine? 

14. Give the two rules for correcting assumed bearings. 

15. Give the rule for correcting old magnetic bearings. 

10. What is the meaning of N 42° E, T M ? S 14° 30' W, M M ? 



CHAPTER VI. 

UNITED STATES RECTANGULAR SURVEYING. 

(101). The statutes of the United States provide that, "The 
public lands shall be divided by north and south lines run accord- 
ing to the true meridian, and by others crossing them at right 
angles, so as to form townships of six miles square, unless where 
the line of an Indian reservation, or of tracts of land heretofore 
surveyed or patented, or the course of navigable rivers may render 
this impracticable ; and in that case this rule must be departed 
from no further than such particular circumstances require." . 

(102). Again, that, "The township shall be divided into sec- 
tions, containing, as nearly as may be, six hundred and forty acres 
each, by running through the same, each way, parallel lines at the 
end of every two miles, and by marking a corner on each of such 
lines at the end of every mile." 

( 103 ). There are many other important provisions relating to 
the subject of Public Land Surveying, but these will suffice. We 
shall now see how they are carried out. 

(104). The fundamental lines upon which a survey is based 
are called the principal meridian and base line. The first of these is 
a meridian of the earth, and the second is a parallel of latitude. 
Their point of intersection is called the initial point. Upon these 
lines every piece of land included in the survey has a direct bear- 
ing, and the whole survey itself is located by the number 'or name 
of its meridian. For instance, the position of a small tract of land 
in Indiana is determined by its distance north or south of the base 
line and east or west of the principal meridian, but the survey of 
nearly the whole State, as well as of other contiguous territory, is 
governed by the second principal meridian which runs north and 
south a short distance west of the center of the State. In like man- 

(60) 



MANUAL OF PLANE SURVEYING. 61 

ner, the survey of Ohio is based upon the first principal meridian 
which serves as the western boundary of the State, the survey of 
Michigan is regulated by the Michigan meridian, and the surveys 
of Minnesota are referred to the fourth and fifth principal meri- 
•dians. 

105 ). The selection of an initial point is the first step in the 
survey of any new territory, and this is always chosen at some 
natural and imperishable land-mark found in or near the lands to 
be surveyed. From this point the principal meridian is surveyed 
north or south, or north and south, and the base line east or west, 
or east and west. Upon these- lines, which are surveyed with a 
fine instrument and with the greatest possible precision, six-miles 
distances are marked for township corners, one-mile distances for 
section corners, and half-mile distances for quarter section corners. 
Each of these corners is marked with a suitable monument, and 
appropriate witnesses are also chosen. 

( 106 ). From each six-miles point on the base-line, east and 
west of the initial point, other meridians are surveyed, and the 
territory is thus divided into strips, each six miles wide, lying 
north and south. These strips, when divided into townships, are 
called ranges. The first one east of the principal meridian is called 
range 1 east, the second is called range 2 east, the third range 3 
east, and so on. In the same way the first one west of the meridian 
is called range I west, the second range 2 west, etc. 

(107). Similarly, lines running east and west from the six- 
miles points on the principal meridian, divide the territory into 
strips, each six miles wide, lying east and west. The meridians 
running north and south and the parallels running east and west 
thus divide the territory into townships, each of which is about six 
miles square, and consequently contains thirty-six square miles 
or sections. The first township north of the base-line in each 
range is called township 1 north, the second township 2 north, and 
so on ; and those south are named 1 south, 2 south, etc., to the 
limit of the survey. The first township north of the base-line and 
east of the principal meridian is described as township 1 north, 
range 1 east, and, in a similar manner, every township is named 
with regard to its distance from the base-line and from the princi- 
pal meridian. This is conveniently shown in Fig. 15. 

( 108 ). Since meridians converge as they approach the pole, it 
is evident that townships can not be quite square, and that every 



62 



MANUAL OF PLANE SURVEYING. 



township must be somewhat smaller than the township south of it 
and larger than the one north of it, except in certain cases on the 
base-line. In the northern part of the United States this conver- 
gence is greater than in the southern part, and the north line of & 




Fig. 15. 



township in some places is more than one hundred feet shorter 
than the south line. To keep the error arising from this conver- 
gence within reasonable bounds, lines called " correction lines" 
are now surveyed every twenty-four miles or four townships on the 
north side of the base-line, and every thirty miles or five town- 
ships south of the base line, and always parallel to it. Upon 
these correction parallels, the distances are measured off anew, 
same as on the base-line, and they become secondary bases in the 



MANUAL OF PLANE SURVEYING. 63 

survey, although townships are all referred to the base-line, just 
as if they did not exist. 

(109). For convenience, auxiliary meridians are also estab- 
lished every eight ranges or forty-eight miles east and west of the 
principal meridian, and the territory is, in consequence, divided 
into rectangles each 48 miles long by 24 miles wide north of the 
base-line, and 48 miles long by 30 miles wide south of the base-line. 

(110). The manner in which the townships are surveyed may 
easily be explained by Fig. 16. Those north of the base-line and 
east of the principal meridian are surveyed by commencing at 
the southeast corner of township 1 north, range 1 east, and run- 
ning north, establishing section and quarter-section corners at 
proper distances, four hundred and eighty chains, or six miles, to 
the northeast corner of the township. From this point the sur- 
veyor runs west six miles, or four hundred and eighty chains, to 
the principal meridian and finishes the survey of the first town- 
ship. He then surveys the next township north in exactly the 
same way, and continues, as indicated by the numbers, until he 
closes the first tier of townships on the correction parallel. In the 
same way he begins at the base-line and surveys the next tier east. 
The townships south of the base-line are surveyed in the same 
way, except that the surveyor works toward the base-line instead of 
from it, as Avhen north, as shown by the numbers. 

(111). West of the principal meridian and north of the base- 
line the survey begins at the southwest corner of township 1 north, 
range 1 west, and proceeds in the order of the numbers. South of 
the base-line the process is entirely similar. It will be observed 
that townships east of the principal meridian are surveyed by run- 
ning first north and then west, and those west by running first north 
and then east. 

( 112 ). Excesses and deficiencies in the length of township lines 
are thrown on the north and west sides of the townships so as to 
fall ultimately into the north and west tiers of sections. These 
are called fractional sections, and will be considered in due time. 

( 113). The township (congressional township) which we have 
had under consideration must not be confounded with the civil 
township. The former is always, when not fractional, six miles 
square, but the latter may be any reasonable size or shape whatever. 

( 114 ). Let us now observe how the townships are divided into 
sections. 



64 



MANUAL OF PLANE SURVEYING. 




Fig. 16. 



MANUAL OF PLANE SURVEYING. 



65 



1. Before the surveyor attempts to make this division, he ascer- 
tains the bearings of the boundary lines of the township, so as to 
be able to run the interior lines as nearly parallel to them as pos- 
sible. 

2. These bearings may generally be easily obtained from the 
notes of the previous survey, according to the directions we have 
already given, but he may also obtain them by retracing a section 
of one or more of the exterior lines of the township and determin- 
ing the bearing from the line itself. He should also measure sec- 
tions of the township line to see how his chain agrees in length 
with the chain used in the previous survey. Having attended to 
these preliminaries, he is ready to begin work. 

3. The townships, as we have before stated, contain thirty-six 
sections each, and these sections are numbered, as shown in Fig. 17. 





90 r : 63 J/7 34 17 ; ; 

6 5 4 3 2 1 

88 , 89 67 50 33 /6 

86 67 65 ... 66 48 49-3' - 32 /4 15 

7 8 9 10-11 12 

34 ,7 3 5 : 64 4-7 30 - 13 

8Z 83 62 631-5 '46 28 29 1 1 It 

10 17 16 15 .14 13 

So ^ . * 8/ -< 6/ •' - - 44 '.".-- '■-'-. 27 ' ■■■■10 7- 

18 79 59 '60 42 A3 25 - 26 8 9 




1<9 20 21 - 22 23 24 

16 17 " 58 4/ 24 7 - 




■■'. .; : 74 775 36 ..;. 57 .39 _ . 40 22 . ,23 5 ; - 6 

30 29 28 21 26 25 

7i " 73 ' 55 ' - 38 2A * 4 

: 70 7; 7/ 5-3 54 36 37 f9 7:20 2. . 3 

31 32 . 33 34 35 36 ;:: 

69 77 -52 7/ 35 : ~" :; . 13 '■• / 





Fig. 17. 



66 ■* MANUAL OF PLANE SURVEYING. 

The manner in which the sections are surveyed will now be ex- 
plained. 

4. The surveyor goes to the south-west corner of section thirty- 
six to begin the survey. At this point he sets his compass at the 
bearing of the east line of the township, which should be, but sel- 
dom is, the true meridian, and runs north forty chains. Here he 
establishes a quarter-section corner between sections thirty-five 
and thirty-six, and then proceeds forty chains or one-half mile 
further to the corner of the section, or rather to the corner of sec- 
tions twenty -five, twenty-six, thirty-five and thirty-six, since these 
four sections have a common corner here. Distances from the 
starting point at which brooks, creeks and other objects of im- 
portance are met on the line, are carefully noted. From this cor- 
ner he runs a random line to the east line of the township. If 
this line intersects the township line at the first mile corner, it is 
marked back as the true north line of section thirty-six, but if it 
does not, the distance which it misses the corner, either right or 
left, is noted, and the line changed accordingly. 

5. Having returned to the north-west corner of section thirty- 
six, he next proceeds to survey section twenty-five in the same 
manner, and he follows the route indicated by the figures until he 
completes the survey of the eastern tier of sections. It will be ob- 
served that when he completes the survey of section twelve, he 
then finishes up the survey of section one by running north on a 
random line and correcting back to the south-west quarter. 

6. He next surveys the second tier of sections by beginning at 
the south-west corner of section thirty-five, and proceeding north 
in the same manner as in the survey of the first tier, and thus he 
continues until he reaches the fifth tier. Here, after surveying 
section thirty-two, he runs west from its north-west corner to the 
range line or meridian and completes the survey of section thirty- 
one, and continues to work north, surveying the fifth and sixth 
tiers together, until he reaches the north line of the township. 

(115). The township is now divided into sections, each of 
which, except those in the north and west tiers, called fractional 
sections, is sold as containing six hundred and forty acres of land, 
more or less, and each quarter-section as containing one hundred 
and sixty acres, more or less. The fractional sections generally 
contain a greater or less quantity of land than the other sections, 
because all excesses and deficiencies fall to them. In surveying 



MANUAL OF PLANE SURVEYING. 67 

them, however, the quarter-section corners between them are so 
placed that the excesses and deficiencies fall to the exterior quar- 
ters, and the interior quarters, or those touching the other sections 
of the township, are sold as containing the proper amount of land — 
one hundred and sixty acres each. The exterior quarters are sold 
as containing whatever the measurements of the survey indicate 
that they contain. Section six is sometimes called the " double 
fractional," and usually contains only one exact quarter. 

(116). Whenever, in the course of a survey, an impassable bar- 
rier, such as a lake or navigable river, is met, the surveyor estab- 
lishes what is called a meander corner on its margin, and then 
runs a meander line from this corner along the edge of the obsta- 
cle. The rivers and lakes thus meandered are reserved in the sale 
of the public lands. 

(117). The surveyor, from the time the survey of the principal 
meridian is begun until the township is divided into sections, 
marks every half-mile of true line that he surveys with a corner, 
and keeps an account in his field -notes of every important object 
he meets in the survey, as well as a topographical description of 
the country. He also makes two sets of corners on the correction 
parallels, one for townships north, and the other for townships 
south of the line. Aside from this it is. not unusual to find two 
sets of corners on interior parallels and meridians, owing to dis- 
crepancies between contiguous surveys. 

(118). The monuments used in marking corners are always 
adapted to the country in which the survey is made, and their 
position is generally witnessed by one or more bearing trees, or 
mounds of earth thrown up around a stake or a stone, whose 
courses and distances from the corner are carefully noted in the 
field-books of the survey. These witnesses, if trees, are always 
marked facing the corner, which enables them to be more easily 
found at any subsequent time. 

(119). This completes the work of the Government Surveyor, 
or deputy, as he is usually called. He returns his notes to the 
Surveyor General of his district, and these notes become the basis 
of all subsequent surveys. The work of dividing and sub-dividing 
the sections, which belongs to the county and private surveyors, 
we shall consider in the next chapter. 

(120). This beautiful system of land surveying, not unlike the 
old Eoman system, was devised about the year 1785, for the pur- 



t)8 MANUAL OF PLANE SURVEYING. 

pose of preparing the North West Territory for settlement, and 
has answered the purpose in an admirable manner. It is the re- 
sult of mature deliberation, and exhibits no mean knowledge of 
engineering skill, and, like many other great inventions, its beauty 
and utility consist in its extreme simplicity. It has long since 
outgrown the limits for which it was intended, and soon nearly 
the whole territory between the western boundary of Pennsylvania 
and the Pacific ocean will be united in one complete net-work of 
sections. 

The readiness with which it enables a surveyor to re-trace old 
lines and determine the location of lost corners prevents an end- 
less amount of litigation common to States not surveyed accord- 
ing to this system. 

Questions on Chapter VI. 

1. What are the fundamental lines of a survey? 

2. What is their point of intersection called? 

3. Upon what principal meridian is the survey of Indiana 

based ? 

4. What is a range? A township? 

5. Draw a diagram representing town 4 south, range 3 east. 

6. How is the error caused by the convergence of the merid- 

ians arrested? 

7. How are townships north of the base-line and east of the 

principal meridian surveyed ? South of the base-line and 
west of the principal meridian? 

8. What is the difference between a congressional township 

and a civil township? 

9. How may a surveyor ascertain the bearing of the lines that 

bound a township? 

10. How many sections in a township? How are they num- 

bered? 

11. Where does a surveyor begin work when he divides a town- 

ship into sections ? 

12. Describe the method of surveying the eastern tier of sec- 

tions. The western. 

13. Name the fractional sections. The double-fractional section. 

14. What causes fractional sections? 

15. How many acres in a section? 



MANUAL OF PLANE SURVEYING. 69* 

16. What quarters of fractional sections are generally full? 

17. What is a meander line? 

18. Why are two sets of corners needed on correction lines? 

19. How far apart are the corners on lines surveyed by the gov- 

ernment surveyor? 

20. For what purpose was the rectangular system devised? 

W T hen? 



CHAPTER VII. 

THE DIVISION AND SUB-DIVISION OF THE SECTION. 

( 121). The county or other surveyor makes all his surveys in 
accordance with the field-notes of the original survey, a copy of 
which form a part of the public records of each county. 

Subsequent surveys may prove that great irregularities exist in 
the original survey, but none of the lines or corners can be changed. 

( 122 ). The idzal section of the young surveyor frequently 



2 


9 


8 


G 


7 


a 


1 


£ 


? 



Fig. 18. 



differs greatly from the real section he meets in practice. The 
ideal section is very nearly a perfect square — varying a little on 
account of the convergence of the meridians ; it is bounded by 
four straight lines, and contains almost exactly 640 acres of land. 

(70) 



MANUAL OF PLANE SURVEYING. 71 

The real section may sometimes be far from square; its boundary 
lines may deflect every half-mile on its perimeter, and its area 
may exceed by several acres the area of another section adjoining. 
The surveyor, however, must 'adhere as closely as possible to the 
original survey, and let the section and divisions and sub-divisions 
of the section contain whatever the government deputy saw proper 
to put into them. 

(123). Let us now see what the principal divisions and sub- 
divisions of the section are, and then consider the method of sur- 
veying them, locating the corners, etc. 

(124). Fig. 18 represents a section, with the main divisions 
and sub-divisions laid off on its face. They are described as fol- 
lows : 

1. S. W. qr. 

2. N. W. qr. 

3. S. i N. E. qr. 

4. E. JS. E. qr. 

5. W. i S. E. qr. 

6. N. E. i N. E. qr. 

7. S. i N. W. i N. E. qr. 

8. N.E.lN.W.JN.E.qi. 

9. N. W. J N. W. i N. E. qr. 

Of these tracts, Nos. 1 and 2 each contain 160 acres ; 3, 4 and 5 
each 80 acres ; 6, 40 acres ; 7, 20 acres ; and 8 and 9 each 10 acres 

( 125 ). Whenever an instrument of writing, such as a deed or 
mortgage, implying responsibility to the amount of the descrip- 
tion, bears on a tract of land, the area -is usually qualified by the 
compound term " more or less." For instance, ISo. 1, above, would 
be described as containing 160 acres, more or less ; No, 3, as con- 
taining 80 acres, more or less, and so on. 

( 126 ). The corners of the section, or of the different parts of it, 
are named from their position. The principal ones are the sec- 
tion corners, J corners, J J corners, and J | corners. Each corner, 
no matter to what class it belongs, is named from its office and 



72 



MANUAL OF PLANE SURVEYING. 



situation. The numbers on the diagram correspond with the 
names after similar numbers below. 



fO 



// 



Tl 



n 



2 


2 I 


8 2 


3 


2 


1 3 


1 


3 


2 


S 2 


2 


ti 













IS 


7 /¥ 






Fig. 19. 


1. 


Section Corners : 




(!)• 


N. E. Corner. 




(2). 


S. E. " 




(3). 


S. W. " 




(4). 


N. W. " 


2. 


Quarter-Section Corner 




(5). 


N. 1 Corner. 




(6). 


E. i 




(7). 


s. i « 




(8). 


W.J " 




(9). 


Center " 



12 



13 



MANUAL OF PLANE SURVEYING. 73 

3. Half-Quarter Corners : 

(10). N. i i W. Corner. 

(11). N. } i E. << 

(12). E. H N. " 

(13). E. its. " 

(14). S. JJE. " 

(15). s. hw. " 

(16). W.HS. 

(17). W.ilNi " 

(18). N. U 

(19). E. § J 

(20). s. u 

(21). W.Ji 

4. Fourth-Quarter Corners: 

(22). N.W. Center Corner. 

(23). N. E. " " 

(24). S. E. 

(25). S. W. " 

( 127 ). The exterior lines of the section are called section lines, 
and the two lines that cross at the center of the section are called 
center lines. 

( 128 ). We are now ready to examine the method by which the 
position of each of the various classes of corners is determined. 

1. The section corners, and the exterior quarter-section corners, 
except in an occasional case on a town or range line, are set by the 
Government deputy, and the surveyor who follows him sets the re- 
maining quarter-section corner (center) and all the minor corners. 

2. The section is divided and corners set according to instruc- 
tions from the proper authority, and with regard to setting the 
center corner of the section, two methods seem to be in use : 

(1). A line is run connecting the N. \ corner with the S. J cor- 



. 74 MANUAL OF PLANE SURVEYING. 

ner, and another connecting the E. J corner with the W. J corner. 
The point of intersection of these two lines is taken for the center 
of the section. 

This is the method in general use, and is perhaps the most equi- 
table one that could be devised : * 

(2). A line is surveyed connecting the E. 1 corner with the W. 
J corner, and the middle point of this line is taken for the center 
of the section. 

If the section lines were straight from one corner of the section 
to the other, the corner determined by this method would coincide 
in position with the one determined by the other ; but as this is 
not always the case, they may differ considerably in position. 

3. To Set a Half- Quarter Corner. — Run a line along the quarter- 
section^ on the side upon which the corner is to be set, and from 
one corner to the other. Bisect this line for the corner. 

For example : To set the E. J J corner, we connect the center 
corner and E. | corner with a line, and the middle point of this 
line is the required corner. 

4. To Set a Fourth- Quarter Comer. -First set a J | corner on each 
of two opposite sides of the quarter-section. Then connect these 
two corners with a line, and bisect this line for the J J corner. 

To illustrate : In order to set the N. E. center corner, it is nec- 
essary first to set the N. J J E. corner and the E. J J corner, or the 
E. J J N. corner and the N. J J corner. The middle point of a 
line connecting one corner with the other, in either set, for it is 
immaterial which is taken, will be the required corner. 

5. The same methods are employed for setting corners to the 
divisions of the fourth of quarter-sections. For instance, to set 
the north-east corner of the S. J N. W. J N. E. qr., it is necessary 
only to bisect the line between the N. E. center corner and N. J | 
E. corner. 

EXAMPLES. 

( 129 ). (1). How would you set the S. J J corner? 
(3). How is the S. J | W. corner set? 
(4). Describe the method of setting the S. W. center 
corner. 

( 130). We are now prepared to understand how the divisions 
and sub-divisions of the section are surveyed. 

1. A quarter-section is surveyed by running the two exterior 



MANUAL OF PLANE SURVEYING. 75 

half-mile lines, and the two center lines of the section, in order to 
locate its interior corner and fix its interior boundaries. * 

2. A half-quarter is surveyed by first surveying as many of the 
lines of the quarter as enter wholly or in part into its boundaries, 
then whatever other lines are necessary to determine its remain- 
ing corners, and finally the lines connecting these corners with 
one another, or with others. 

For instance, to survey the S. J S. E. qr., it would be necessary 
to run the south line of the quarter, the east line, the two center 
lines of the section, and the E. and W. center line of the south- 
east quarter. 

3. A fourth-quarter is surveyed on the same principle as the 
half-quarter. 

For example, in surveying the S. E. J S. E. qr., it is necessary 
to survey the south and east lines of the quarter, the E. and W. 
and N. and S. center lines of the section, either the E. and W. or 
the N. and S. center line of the S. E. qr., and finally the north line 
of the fourth-quarter, if the N. and S. center line has been sur- 
veyed, or the west line, if the E. and W. center line has been sur- 
veyed. 

( 131 ). As a general rule for the survey of a tract of land bear- 
ing a relation to the section, the following is submitted : Kun 
lines to connect the known corners of the tract when no unknown 
corner intervenes between them, then such lines as are necessary 
to determine the location of unknown corners, and finally lines to 
connect these newly located corners with one another, or with 
others. 

( 132 ). In many cases some, if not quite all, the boundary lines 
of a tract of land are known. There is seldom any need of re- 
surveying these, except where it must be done in order to establish 
other lines or corners. 

EXAMPLES. 

(133). What lines would a surveyor have to run in order to 
establish all the boundaries of each of the following described 
tracts? 

(1). S. W. qr. 

(2). N. i N. E. qr. 

(3). S. i N. W. qr. 

(4). N. E. J S. W. qr. 

(5). N.W.JN.W.qr. 



76 MANUAL OF PLANE SURVEYING. 

(134). So far we have dealt exclusively with corners, line-, 
and tracts which may be called independent: 

The corners, because their position is determined by the division 
of certain lines, and is not definitely fixed, so far as distance is 
concerned, by any other point in the section. 

The lines, because they connect the corners, and may, therefore, 
vary in a limited degree either in course or distance, or both. 

The tracts, because they are limited by the lines and are not 
definitely fixed as to area or figure. 

(135). To illustrate the preceding still further, suppose the 
north line of a quarter-section to be 36 chains long, instead of 40 
chains long, and the contents of the quarter-section to be 148 
acres, instead of 160 acres. The J J corner on the north line will 
then be 18 chains from the section corner, and the same distance 
from the J corner, and each of the lines will be but 18 chains in 
length, and a fourth-quarter out of the quarter-section may fall 
short three or four acres. If, now, the north line of the quarter- 
section had been longer, the N. J J corner would have been 
further from each of its north corners, and the area of the 
fourth-quarter would have been greater. 

(136). There is a class of corners, lines, and tracts, however, 
that may be called dependent: 

The corners, because their distance is fixed from some given 
point, and is not obtained by bisecting a line. 

The lines, because they connect the corners, and are conse- 
quently of a definite course and distance. 

The tracts, because they are bounded by the lines, and their 
area, therefore, is not affected by the excess or deficiency of land 
in the section. 

Dependent corners, then, are those whose position is definitely 
fixed; dependent lines connect dependent corners, and dependent 
tracts are bounded by dependent lines. 

(137). Dependent lines and tracts are surveyed without any 
reference whatever to the division or sub-division of the section, 
although they may depend on some corner in the section as a base. 

A tract may be partly dependent and partly independent. 

( 138). The following are examples of descriptions of depend- 
ent tracts : 

(1). N. 45° E. 16.00; thence N. 45° W. 10.00; thence S. 45° W: 
16.00; thence 8/45° E. 10.00, to the place of beginning. 



MANUAL OF PLANE SURVEYING. 77 

(2). Commencing at the S. E. corner of section 22, town 3 N., 
range 4 E. and running thence N. 15° E. 12.00; thence S. 45° E. 
12.00; thence S. 75° W. 12.00, to the place of beginning. 

( 139). In a dependent tract the course and distance of each of 
its boundary lines are usually given in the description of it, and 
it is then said to be described by " metes and bounds." 

(140). The rules given for setting corners, running lines, and 
surveying tracts, in full sections, also apply to fractional sections, 
except where their fractional sides do not contain half, or much 
more than half, the usual amount of land found in these parts 
of- sections, or where the amount they contain considerably ex- 
ceeds the usual amount. In the first case, the outside tier of 
fourths- of-quarter is omitted, and in the second the excess is 
usually thrown to them, and the inside tier of fourths-of -quarter 
in the outside quarters of the sections, are left of their usual size. 
If the deficiency is very great, perhaps the entire outside tier of 
quarters is wanting. 

( 141 ). When a tract of land lies partly in one quarter of a 
section and partly in another, each part should generally be de- 
scribed and surveyed separately ; and the same may be said of 
tracts extending into two or more sections, and sometimes also of 
those extending into different fourths of the same quarter. The 
following are examples : 

(1). N. | N. E. qr., and N. E. | N. W. qr. 

(2). S. W. i N. E. qr., and N. £ N. W. | S. E. qr 

(3). N. E. qr. Sec. 4, and N. W. qr. Sec. 3. 

(4). S. E. J S. E. qr. Sec. 35, and W. } S. W. qr. Sec. 36. 

(5). S. E. i N. W. qr., and E. J N. E. J N. W. qr. 

In some of the cases an occasional line may be a boundary to 
each part of the tract, as the east line of No. 5, described above. 

Questions on Chapter VII. 

1. Does the section always contain exactly 640 acres? 

2. Draw a diagram of a section and represent the following 

tracts on it: N. E. qr.; N. W. \ N. W. qr.; S J S. W. qr.; 
N. i N. E. i N. W. qr. 

3. Why is the phrase " more or less " used in descriptions of 

land in deeds, etc.? 



8 MANUAL OF PLANE SURVEYING. 

4. Name the quarter-section corners. The J \ corners. The- 

J J corners. 

5. What are the " center lines " ? 

6. What eight corners does the Government deputy establish 

to nearly every section ? 

7. Does he ever set an interior corner of a section? 

8. Give the first method of finding the center of a section? 

9. Prove that the first and second methods would agree, if the 

section lines were straight from one section corner to 
another. 

10. How do you set a J } corner? A \ \ corner? 

11. How would you survey a quarter section ? A half-quarter? 

A fourth-quarter? 

12. Give a general rule for the survey of a. tract of land bearing 

a relation to the section. 

13. What is the difference between independent and dependent 

corners ? Lines ? Tracts ? 

14. Describe an independent tract. A dependent tract. 

15. When do the general methods for dividing and sub-dividing- 

sections not hold good in fractional sections ? 

16. When should different parts of a tract be surveyed sepa- 

rately ? 



CHAPTEE VIII. 

FIELD-NOTES. 

( 142). The field-notes of sectional surveys by the Government 
deputy show : 

(1). The witnesses taken at section and quarter-section corners. 

(2). The length of the fractional lines in fractional sections. 

(3). The number of acres in each of the fractional quarters of 
fractional sections. 

(4). The offsets between section corners in one township and the 
corresponding ones in the adjoining township. These offsets are 
sometimes found on town and range lines as well as on correction 
parallels. 

(5). The distances from the starting point of a line at which 
brooks and creeks are crossed, and trees and other objects met with 
on the line. 

(6). A description of the timber, surface, soil, etc 

(7). The courses and distances of meander lines surveyed along 
rivers, lakes, etc. 

(143). Each full section is supposed to contain 640 acres of 
land, and it is always taken for granted that the distance between 
a section corner and a quarter-section corner is 40 chains. The 
lines are also supposed to run due north and south and east and 
west. These suppositions, however, are strictly correct only in 
comparatively few cases. 

Quarter-section corners, like section corners, on town and range 
lines answer for sections on each side of the line, but where an 
offset occurs aud the closing section corner is set either at one side 
or the other of the corner already on the line, the quarter-section 
corner for the closing section is omitted. In this case the closing 
section has only seven corners instead of eight. 

(79) 



80 



MANUAL OF PLANE SURVEYING. 



(144). For convenience of reference a plot of the township is 
drawn after the survey is completed, and whatever is essential to 
subsequent surveys is represented on it. 

( 145 ). Fig. 20 will give an idea of the manner in which a plot 
of this kind is drawn, although space will not permit its being 
made as complete as it should be. 



G / T e J£ d 1> J G i B.a A 



S 

s 



i 

T 

J 
K 



M 



6 cj 5 o 4 c> 3 



18 



19 



30 



, 8 in 9 



17 -*■ 16 '*■ 15 'f 14 -* 13 



nr> 



20 



C4 



29 



C) 



cp 



21 



rsi 



28 



[0 l« !! i, 12 



22 



(j 



27 



31 -- 32 -- 33 -- 34 -- 35 -- 36 



23 



26 



24 



cj 



25 



x 

X 

w 

w 

V 

u 

u 

t 
T 



m JH n o T p Q # JR * S 

Fig. 20. 



The capital letters on the margin designate the section corners 
on the town and range lines, and the small letters the quarter- 
section corners. 

The interior section corners are designated by the numbers of 
sections. Thus, the southwest corner of section one is numbered 



MANUAL OF PLANE SURVEYING. . 81 

1, 2, 11, 12, because it serves as a corner to each of these sections. 

The interior quarter-section corners are designated as corners 1 
to 6, respectively, on the line BR,I W, or whatever line it may be. 

( 146 ). Attached to the plot of the township is a list of the wit- 
nesses at each of the corners, generally on the principle of the fol- 
lowing : 

1. Exterior corners. 

Sections. 

A. Be 6 N 14 W 32, Ash 16 S 12 W 17. 

B. Oak 10 S 19 E 11, Hickory 18 N 5 W 6. 

C. Maple 14 N 12 E 41, Poplar 28 S 72 W 19. 
Quarter-sections. 

a. Oak 14 N 61 W 14, Sugar 15 S 16 W 13. 

b. Elm 26 S 12 E 25, Ash 9 S 63 W 14. 

2. Interior corners. 

Sections. 

1. Cor. of 1, 2, 11, 12, Maple 15 N 10 E 19, Elm 16 S 71 E 12. 

2. Cor. of 2, 3, 10, 11, Oak 36 S 15 W 12, Ash 20 82 W 41. 
On the line B B, (Quarter-section corners). 

(a). At 1. Be 12 N 16 E 42, Gum 14 S 15 W 22. 
(b). At 2. Pop. 28 S. 46 E 27, Ash 20 N 29 W 31. 

In a similar manner the quarter-section corners on the lines H 
X, I W, J V, etc., are also numbered and the witnesses given to 
each. 

Each of these lists is extended so as to include all the corners of 
that particular class. 

t ( 147 ). The following particulars are usually shown on the face 
of the plot: 

Length of fractional lines : 

B to 6, 39.07. 

C to 6, 38.49. 

D to 6, 38.03, and so on with all the other fractional lines. 
6 



82 . MANUAL OF PLANE SURVEYING, 

1. Area of fractional quarters. 

N. E. qr. sec. 1, 159.17 acres. 
N. W. qr. sec. 1, 157.51 acres. 
N. E. qr. sec. 2, 155.70 acres. 

2. Creeks, etc. 

N from E 31.42, creek running S. W., 43 links wide. 

E from 25, 26, 35, 36, 22.16, creek running S.W., 41 links wide, 

3. Offsets. 

B, 41 links E. of corner in town north. 

L, 59 links S. of corner in range west. 

Only a few instances are cited in each case to show the general 
plan. 

SUBSEQUENT NOTES. 

( 148 ). Every surveyor should be provided with a copy of the 
original field-notes of his county, together with notes of all the 
surveys made by his predecessors. These notes, with the addi- 
tions he himself makes from time to time (provided he and his 
predecessors are authorized surveyors), constitute the surveyor's 
records of the county. 

(149). These records should contain a plot of each piece of 
land surveyed, showing its area and the course and distance of 
each of its boundary lines. The manner of drawing these plots is 
explained in the chapter on " Plotting." 

( 150 ). From the regular county record is usually drawn a 
pocket record for field use (1) of the original survey, and (2) of the 
subsequent surveys. The notes of the original survey generally 
fill but a small book, and may be arranged according to the 
method given above ; but, unless the county is unusually small, it 
is more convenient to have a separate book for each range in 
which to enter the notes of the subsequent surveys. 

( 151 ). Each of these books should contain at least twice as 
many pages as there are sections in the range. The left-hand 
page in each one should contain a plot of a section about 4 inches 
square divided into quarters, and the right-hand or opposite page 
should be left blank, so that notes of the successive surveys in the 



MANUAL OF PLANE SURVEYING. 



sa 



section may be entered upon it. These refer to the plot by num- 
bers or letters in the manner shown in the figure. 

1. (Left-hand page). 

Section , 

Town , Eange 

i 



li 



/ 





V0.20 


C5 


CL 


$ 


§ 


5a 


^ 
^ 




S 
^ 






10. /s 




* 




5l 




e *^. 2f 





<k 



Fig. 21. 

(This plat is only one-fourth the size suggested above, but will 
serve to illustrate). 

2. (Eight-hand page). 

A. Be 16 N 12 E 19, Ash 14 S 6 E 12. 

B. Pop 28 S 60 W 19, Walnut 30 N 16 E 41. 

C. Be 13 S 16 E 17, Elm 18 N 22 E 31. 

D. Pop 16 N 19 E 27, Ash 18 N 23 W 21. 
a. Be 23 S 17 W 40, Elm 16 N 21 W 14. 

Bearings of principal lines : 
AF.N 89° 35 / E. 
C E. N 1° 16' W. 

( 152 ). In these cases the bearings are all given on the basis of 
the true meridian. When the magnetic bearings are given, they 
should be accompanied with the date at which they were taken * 



84 MANUAL OF PLANE SURVEYING. 

(153). All interior surveys in the quarter-sections should be 
represented by proper lines on the plot. The dotted line through 
the north-east quarter in the figure indicates that this quarter has 
been divided into north and south halves. The courses of roads 
and creeks may also be shown on the plot. 

(154). When the bearing of any independent line is not 
known, it may generally be approximated by comparing it with 
other lines in the section or adjoining sections. For instance, 
there is but little difference between the bearing of the line A B 
in the figure and that of the line C D, since the distance between 
their northern extremities differs but 4 links from the distance 
between their southern extremities, and both lines are about of 
the same length. 

(155). The bearing of the line A B may be determined by the 
following method, which has been explained in a preceding chap- 
ter: 

70 : 4 : : 60' : x= 3'+ = amount of correction. 
1° W — 3 / = l° 13' = variation of line A B. 

» (156). This method of determining the bearing of one line 
from that of another depends on the following principles: (1) 
Two parallel lines have the same bearing, and (2) the difference 
in bearing of two lines not parallel is in proportion to their in- 
clination to one another. a 

By reversing this rule we may determine the distance between 
two lines at successive points when their distance apart at one 
point and their difference of bearing are known. 

( 157 ). The surveyor's field-book is the memorandum he keeps 
of his field-work. It contains only the rough entries, which are 
changed in form and transmitted to the records. 

( 158 ). In surveys of independent tracts, perhaps the following 
method of keeping it is as good as any : 

Let us suppose a survey of the south-east quarter of section 9, 
town 8, range 10, commencing at the south-east corner of the sec- 
tion, and made March 29, 1881. If all the corners to the quarter- 
section have been established previously, and the surveyor runs 
the east line first, the entries may be somewhat as follows : 

Mar. 9, 1881. 
9 — 8 — 10. 

Commenced S E cor. and ran N 2° 15' -W, 40.20 to E \ cor., Be 



MANUAL OF PLANE SURVEYING. 8o> 

20 N 15 E 36, Ash 10 N 41 W 12, M E 16 links. At 16.30 from 
S E cor. crossed brook 8 links wide flowing S E. 
. Com. E \ ran S 88° W W, 40.12 to center of section. 19.00 
crossed brook 6 links flowing S E. 

Com. cen. ran S 2° 30' E, 40.20 to S J. 

Com. S i ran N 88° 40" E, 40.15, MLS links. 

The first line terminated 16 links east of the corner, showing- 
that the assumed bearing was about \ degree too small. The 
second and third lines struck the corners, but the fourth ran 5 
links to the north, perhaps on account of some slight error in set- 
ting the compass, as the assumed variation was correct, if we com- 
pare with the second line run. 

(159 ). Wherever the witnesses are found in bad condition new 
ones are taken, as was done at the E \ corner in this survey. 

(160). In dependent surveys it is generally best to have the 
page of the field-book ruled in five vertical columns: The first 
giving the relative name or number of the station or corner at 
which the line begins; the second, its course; the third, its dis- 
tance; the fourth, the number of links missed to the right; the? 
fifth, the number of links missed to the left, as follows : 

Sec. 5, Town 6, Range 4. 

Mar. 18, 1881. 



Sta. 


Course. 


Dis. 


R. 


L. 


A 


N 10° W 


16.00 


4 


1 


B 


N 4°W 


4.00 






C 


N 16° 30' E 


18.00 




6 


D 


S 81° 15' W 


9.00 






E 


N 14° W 


11.24 






F 


S 29 c W 


7.26 


5 





Fig. 22. 



( 161 ). The station at which the surveyor begins is called "A," 
and the succeeding ones are named in the order of the letters that 
follow. 

Witnesses taken at any of the corners may be described on the 



$6 MANUAL OF PLANE SURVEYING. 

opposite page and referred to the corner by the proper letter, as 
follows : 

At B. Be 24 N 23 W 16, S E corner house N 81 W 46. 
" E. Elm 22 S 16 E 32, Oak 23 M 12 W 29. 
" F. Large stone at corner. 

The names of all the assistants in the survey are generally re- 
corded, so that they may be known, if evidence should be needed* 
In case of future disputes over the survey. 

( 162). The record made by county and other authorized sur- 
veyers is taken as prima facie evidence in favor of the surveys 
made by them, and particular care should be taken in making 
this record, as well as in field-work, to see that no mistakes are 
committed. 

(163). Other methods of keeping field-notes are also employed 
by surveyors, but the ones described above are perhaps the most 
simple, and they will answer every purpose. 

Questions on Chapter VIII. 

1. What particulars are enumerated in the original field- 

notes ? 

2. By referring to the original notes, how would you find the 

witness to the south-west corner of section 21 ? The W i 
corner of section 9? The S J of section 22? 

3. What sections touch section 15? 29? 26? 

4. What do you understand by " Subsequent Notes?" 

5. What constitute the surveyor's records of a county ? 

6. How is the bearing of an independent line sometimes ap- 

proximated ? 

7. The north line of a quarter-section is 40.32 long, and the 

south line 39.97 long. If the bearing of the east line is 
N 2° 19 W, what is the bearing of the west line? 
-8. Describe the field-book for independent tracts. For de- 
pendent tracts. 
9. When are new witnesses taken to a corner? 
10. What is meant by prima facie evidence? Answer — Evidence 
that establishes a fact, unless set aside by stronger evi- 
dence. 




CHAPTER IX. 

RE-LOCATION OF CORNERS. 

( 164). Perhaps, as a general thing, the most perplexing part 
of a surveyor's work consists in re-locating the corners of the 
original survey. As long as these or the witnesses to them re- 
main, he seldom has any serious trouble in the survey of any 
independent line of the section; but if one of them chance to 
be lost, it must be re-located before any line bearing on it can be 
surveyed, and its re-location, except in certain cases, is frequently 
a matter of difficulty, if not, in some instances, of impossibility. 

In the latter case there is no alternative, except to establish a 
new corner. 

(165). This difficulty arises from the fact before stated, that 
section lines are usually broken at every original corner, and 
these parts, into which the lines are divided, differ from one an- 
other in length, so that the course and distance of one line may 
not be the same as that of any other line in the vicinity, and it 
would not do, therefore, to re-locate a corner by a line having the 
same course and distance of a similar line, either in the same sec- 
tion or any other section. The surveyor, consequently, must 
resort to other means. 

1. In the first place a diligent search should be made for re- 
mains of the monument or witnesses at the missing corner. If 
the witness trees have disappeared, it may be possible to find the 
roots, especially if the ground has not been plowed, as traces of 
them remain for many years. It is probable that some person in 
the vicinity can give him some information relative to the corner 
that will enable him to judge approximately as to where he 
should look for the witnesses. 

2. If, however, all search prove futile, and the course and dis- 

(87) 



88 MANUAL OF PLANE SURVEYING. 

tance of a line connecting this corner with some other corner that 
can be found, be known, the missing corner may sometimes be 
found by running this line from the known corner. 

3. Or if the line run through the woods, it may be possible to 
retrace it by the blazes on the trees, and thus determine the miss- 
ing corner, providing the length of the line be known. 

4. And, again, where subsequent surveys have been made in 
one of the sections touching the corner, some of the subsequent 
corners, taken in connection with the original corners, may enable 
the surveyor to re-locate the missing corner by " projection," as 
follows : 

(1). Suppose the S J- corner of the section to be missing, and 
that the S E corner and S J J E corner can both be found. Now, 
the S J i E corner was evidently set while traces of the S J corner 
remained, and is midway between it and the S E corner. If now 
we begin at the S E corner and survey a line westward, measuring 
to the S \ \ E corner and producing the line an equal distance 
beyond it, the extremity of this line must mark the missing cor- 
ner. If the line run either to one side or the other of the S \ \ E 
corner, the distance to the right or left must be noted. The S \ 
corner will be twice this distance on the same side from the ex- 
tremity of the line, as may be seen by noticing Fig. 23. 

The numbers on the horizontal (true) line indicate the length 
of each section of it, and those on the dotted vertical lines show 
the distance between the random line and the true line at each of 
the corners. 

(2). This is simply the reverse of the method used in setting 
the S \ i E corner. 

(3). If, instead of the S E corner and S \ \ E corner, we have, 
for instance, the S E, E } J, and S E center corners, it will be 
necessary first to project the 8 J \ E from the E \ \ and S E cen- 
ter, and then we may project the S 1, same as before. 

examples : 

(4). 1. If we have the center and S \ \ corners, how may we 
find the S J? 

2. The W J, S \ J, and S W center corners are known, how may 
the S W corner be determined? 

3. The E J, E \ J, and S \ }, corners can be found, how may the 



MANUAL OF PLANE SURVEYING. 



89 



(5). It must be borne in mind that two corners must be found 
on the same side of a quarter-section before the third can be re- 
located by this method. 



16 



JJando 




Fig. 23. 

5. When the surveyor finds it impossible to re-locate a missing 
corner by any of the foregoing methods, or any other method, he 
proceeds to establish a new corner, and in doing this he presumes 
.that the quarter-section lines do no bend at the corner to be established y 
and, if it be a quarter-section corner, that it is midway between the 
corners of the %ection. 



DO MANUAL OF PLANE SURVEYING. 

( 1 ). Suppose the missing corner to be the S J, the new corner 
would be set by bisecting the line connecting the S E and S W cor- 
ners of the section in the same way that a J J corner is set by bi- 
secting the line between the two corners of the quarter-section. 

( 2 ). The corner thus established may be identical with the lost 
corner, or may be some distance from it, but it is the best that can 
be done. 

(3). In the figure a case is illustrated in which the new corner 



V 



/ 



A* A 

1 40.2 1 Sl# 40 - 2* 

Fig. 24. 

is a considerable distance from A, the supposed approximate loca- 
tion of the old corner. The dotted lines represent the old lines, 
and the numbers below the new line show the length of each sec- 
tion of it. Its bearing is marked above it, and in this case indi- 
cates its angle with a line running due east and west. 

(4). If the corner of a section be lost, a new one is set by sur- 
veying the exterior lines of the adjacent quarter-sections as if they 
did not bend at the section corner. The new corner will be at the 
point of intersection of the two lines thus surveyed. 

For instance, to set a new S E corner to section 2, connect the 
8 \ corner of section 1 with the S J corner of section 2, and the E 
| corner of section 2 with the E J corner of section li. The point 
at which the lines cross will be the new corner. 

(5). As in the former case, this corner may not be identical 
with the old corner. Fig. 25 represents a possible case in which 
they are some distance apart. In actual work, however, such ex- 
treme cases as we have noticed will seldom, if ever, come up. 

( 6). In any case, in setting a new section corner, if any \ cor- 
ner can not be found, the line must be produced to the next corner 
that can be found. This may cause one of the lines to be l| or 
even 2 miles long, but the corner is set at the point of intersection, 
same as before. 

(166). In re-locating the original corners to the variable quar- 



MANUAL OF PLANE SURVEYING. 



91 



iers of fractional sections any of the first four methods given above 
may be employed, but when a new corner must be established the 
5th or last does not always hold good, except for the J corner on 
the town or range line, which is set midway between the section 
corners, as in full sections. 

( 167 ). To set the |- corner between two fractional sections, run 



•'JR' CO 



I 



Set. 2. / 



'A cor '^ . 



N i A 'ii 
\ 

Sec, //• \ 
\ 



Ti ' 



S*0« h 



~ . ^ — .— y /4 cor 






'4 ear. 

Fig. 25. 

a line from the interior section corner westward or northward, as 
the case may be, to the exterior section corner on the town or range 
line, and locate the corner 40 chains from the starting point. 

(168). To set an exterior corner to a fractional section, or to 
any exterior section. 

1. If there be an offset between the corner of one section and 
that of the corresponding section in the other town or range, the 



92 MANUAL OF PLANE SURVEYING. 

corner may be re-located by measuring this offset along the town 
or range line, or correction parallel, in the proper direction. 

2. When there is no offset, the corner must be set by crossing 
the lines, according to the method used in interior sections. 

EXAMPLES. 

( 169 ). 1. How may a new E \ corner be set to section 11, pro- 
viding the section corners on that side can both be found? 

2. What must be done, if one or both of the section corners are 
lost, before the quarter-section corner between them can be set? 

3. How do you establish a new interior section corner ? 

4. If one or more of the exterior corners to the adjacent quar- 
ter-sections be lost, what must be done in order to establish the 
section corner ? 

5. How do you establish the -]- corner section between two frac- 
tional sections? 

6. How do you re-locate an exterior section corner when there 
is an offset? on the town or range line ? 

( 170 ). Subsequent corners, whether independent or dependent.. 

are re-located according to the rules by which they were located 

at first, and the same holds good for original corners on meander 

lines. 

Questions on Chapter IX. 

1. When it is found impossible to re-locate a corner, what must 

be done? [corner? 

2. Why is not the new corner always identical with the original 

3. Explain each of the different methods of re-locating original 

corners. 

4. In re-locating an original | corner by "projection," the ran- 

dom line was found to be 16 links to the left of the \ \ 
corner. In what direction and how far will the re-located 
corner be from the extremity of the random line? 

5. How many corners must be found on the side of a quarter- 

section before the remaining one can be re-located ? Why ? 

6. What section in the township north corresponds with section 

2? With section 5? What one in the township west cor- 
responds with 7 ? With 30 ? 

7. What townships touch T 2 N, K 3 E? T4S, K5W? T6 

N, K1W? 

8. How are subsequent corners re-located? 



CHAPTER X. 

DESCRIPTIONS OF LAND. 

(171). No piece of land can be sold or surveyed, unless its de- 
scription is known, and this description should be just as concise 
and simple as possible. 

(172). It would be better, if the length of lines and area of 
tracts were always given in surveyor's measure, instead of in or- 
dinary linear and square measure; yet when this is not done, they 
may be reduced to their equivalents in surveyor's measure by the 
following tables : 

(173). Linear Measure. 

100 links = 1 chain = 4 rods. 
1,000 " =10 " =1 furlong. 
8,000 " =80 « .= lmile. 

Square Measure. 

« 1 sq. chain == 16 sq. rods. 
10 " " = 1 acre. 
6,400 " " = 1 sq. mile. 

It is plain that rods may be reduced to chains by dividing by 4; 
furlongs to chains by multiplying by 10 ; and miles to chains by 
multiplying by 80. 

examples. 

(174). 1. Reduce 15 rods to chains. 

2. Reduce 1 fur. 3 rods to chains. 

3. Reduce 1 mi. 3 fur. 24 rods to chains. 

(175). Fractional parts of a chain should be expressed in 
links : Thus, lOf chains should be written 10 chains and seventy- 
live links, or simply 10. 75. 

(93) 



94 MANUAL OF PLANE SURVEYING. 

(176). As a general rule for reducing from ordinary long or 
linear measure to surveyor's measure, perhaps it would be well to 
use the following : 

Reduce the denominations expressing the length of the line to 
rods, and multiply by .25. The product will be the length of the 
line expressed in chains and links. 

To illustrate, suppose the length of a line to be 7 fur. 16J rods — 
296J rods. = 296.5 rods. This multiplied by .25 equals 74.125, or 
7i chains 12J links. 

(177). The area of tracts in surveyor's measure is always 
given in acres and hundredths, instead of in acres, roods, rods, 
etc., as ordinarily. This will be explained in the chapter on 
Computation of Area. 

(178). Whenever an independent tract is to be described, 
nothing whatever should be said of the course and distance of any 
of its boundary lines, and it should be described simply as such a 
division or sub-division of the section ; as, for instance, the south- 
west quarter, or the north half of the north-east quarter, or the 
north-west fourth of the south-east quarter, or the north half of 
the south-east fourth of the north-west quarter, etc., etc. 

( 179). Errors like the following are frequently made in de- 
scriptions: Forty acres in the form of a square in the south-east 
corner of the section ; eighty acres off the south side of the north- 
east quarter; one hundred and sixty acres in the north-east corner 
of the section ; a strip twenty chains wide off the north side of the 
south-west quarter; and so on. 

Each of these descriptions is faulty, because the independent 
division intended to be described may overrun or fall short in the 
amount of land named in the description. If the tracts were not 
independent, the descriptions would be good. 

Correct the following descriptions : 

( 180 ). 1. 160 acres off west side of section. 

2. Forty acres in the south-west corner of the north-east quarter. 

3. Commencing at the N E cor. of the section ; thence running 
south 20 chains; thence west 40 chains; thence north 20 chains to 
the N J cor. ; thence east to the place of beginning. Containing 
80 acres. 

4. 60 acres off the south side of the south-east quarter. 



MANUAL OF PLANE SURVEYING. 95 

(181 ). Sometimes descriptions contain errors that render them 
worthless. The following are a few examples ; tell where the 
error lies in each one : 

1. NEjNWqr. 

2. SW|NE qr., containing 80 acres. 

3. SJNJNEqr. 

4. 60 acres in N E qr. 

5. N W qr sec. 28, containing 80 acres. 

6. Running north ; thence east 50 chains. 

7. Running S 43° E, 11.21 ; thence N 32° W, 5.26 ; thence 

S 8° W E, 16.32, to the place of beginning. 

Mistakes like the preceding are frequently made by persons who 
are careless, or do not understand how lands should be described,- 
and sometimes give rise to vexatious litigation. 

( 182 ). It is best in nearly all cases to qualify the area of the 
tract described by the phrase " more or less/' as, perhaps, no two 
surveys of the same tract, particularly if it be large, will ex- 
actly coincide throughout, and of course the area will vary with 
the length of the lines. 

(183). In describing dependent tracts the course and distance 
of each of their boundaries should be given, except, perhaps, in 
occasional cases where they have a natural or artificial boundary, 
as for instance, a creek or road, whose course and distance may be 
determined at any time ; but it is always best to be definite in re- 
gard to boundaries when possible. 

( 184). The description should also state whether the bearings 
are based on the true meridian or on the magnetic meridian. If 
based on the magnetic meridian, the date at which they were 
taken should be given. 

( 185 ). Where a line is described as running north, a due north 
and south line is meant, and the same is true of south. Similarly, 
an east line means one running due east, and a west line one 
running due west. 

( 186). The survey of a tract of land is always made in ac- 
cordance with the description, except where an obvious mistake 
occurs, in which case the surveyor will have to exercise his judg- 
ment in regard to the course to be pursued, as no rule can be given 
that will apply to all cases. However, the decisions in the "Ap- 
pendix " may assist him somewhat in arriving at a conclusion. 



96 MANUAL OF PLANE SURVEYING. 

Sometimes the mistake is made in writing the original descrip- 
tion of the tract, and at others in copying from preceding titles 
and deeds. In the latter case, a comparison of the deeds will show 
in what it consists. As soon as a mistake is discovered in a deed 
or mortgage, or in any other instrument in which" a great deal 
may depend upon the description, steps should be taken by those 
interested to have it corrected. 

None but competent persons should be chosen to write descrip- 
tions of land. 

Questions on Chapter X. 

1. Why can not a tract of land be sold or surveyed without a 
description? 

2. How many links in a rod? Chains in a mile? 

3. Write 17 chains and 46} links decimally. 

4. Give the general rule for reducing from ordinary long or 

linear measure to surveyor's measure. 

5. How is the area of a tract of land expressed in surveyor's 

measure? 

6. Why should not the metes and bounds of an independent 

tract be given in a description ? 

7. Why should the phrase " more or less " be inserted in a de- 

scription ? 

8. Why is it necessary to state whether the bearings are based 

on the true meridian or on the magnetic meridian? 
• 9. If on the magnetic meridian, why should the date at which 
they were taken be given ? 



CHAPTER XI. 



OBSTACLES TO ALIGNMENT AND MEASUREMENT. 

( 187 ).' It frequently happens in the course of a survey that the 
line strikes an obstacle of some kind — as, for instance, a building, 
or a large pond, or creek — that obstructs the measurement, if not 
both line of sight and measurement. 

( 188 ). These obstacles may be divided into two classes : (1) Ob- 
stacles that may be spanned by measurements along their sides 
or margins, as a building, a pond, etc. ; (2) obstacles that can not 
be spanned in this way, as rivers and lakes. 

( 189). Various methods are employed for spanning obstacles, 
but only a few will be given, in order to prevent confusion. 

First Class of Obstacles. 
1. By Perpendiculars. — Fig. 26 represents an obstacle on the line 

C D 




Fig. 26. 

A B which runs nearly to the side of it. At the extremity B, a 
perpendicular, B C, is measured long enough to permit the line C 
I> to pass the obstacle. In this case the perpendicular is fifty links 
long. The line C D is then run at the bearing of the line A B, and 
is, consequently, parallel to it. From the extremity, D, of this 
line another perpendicular, D E, of the same length as the first, 
7 (97) 



MANUAL OF PLANE SURVEYING. 

is measured, which, of course, terminates on the original line pro- 
duced through the obstacle. The survey of the line may then be 
continued from E in the direction E F at pleasure, and the length 
of C D added to the regular sections, A B and E F. 

2. By an Equilateral Triangle. — The line A B terminates some- 
what further from the side of the obstacle than before, and the 



A. 




E 



Fig. 27. 

line B C is then laid off at an angle of 60° with the. line A B pro- 
duced and measured to a suitable distance. In the case before us 
it is 1 chain and 25 links in length. From the extremity, C, of 
this line, the line C D, of equal length with it, is surveyed at an 
angle of 60° with C B. We then have an equilateral triangle, and 
the side B D is also 1 chain and 25 links in length. The line may 
then be continued, and the distance through the obstacle, 1 chain 
and 25 links, added to the other sections, as before. 

3. (a) By a Right-angled Triangle. — This method is similar to 
the preceding one, and differs from it only in having a right-angle 




Fig. 28. 



at C, and angles of 45° at B and D in the triangle used. The side 
B C is first surveyed, and then C D at right-angles to it and of 



MANUAL OF PLANE SURVEYING. 



99 



equal length. The distance from B to D is found by extracting 
the square root of the sum of the squares of B C and C D, as the 
side B D is the hypothenuse of the triangle. In this case the dis- 
tance from B to D is equal to y^lOO) 2 + (100) 2 = 1.414. 

(b) When the obstacle is a pond, or something that does not 
obstruct the line of sight, the following method will be found most 
convenient : 

c 



A 




E 



Fig. 29. 

The line is measured to B, near the margin of the pond, and the* 
flag set at D on its continuation on the opposite side. C B is then? 
measured perpendicular to A B, and lastly the line C D is meas- 
ured. We have now a right-angled triangle whose base is required' 
and may be found by extracting the square root of the difference 
between the squares of CD and B C. In this example the base- 
B D equals ^(Ud) 2 — (75) 2 = 1.00. 

In every case the line is to be continued from D at the bearing 
of the first section, A B, and the distance through the obstacle- 
must be added. 

4. By Symmetrical Triangles. — When, as in the last case, the line 
of sight is not obstructed, the following method may sometimes^ 
be used : 








Fig. 30. 



100 MANUAL OF PLANE SURVEYING. 

From the extremity, B, of the line A B measure a line to C and 
produce it to F, an equal distance beyond, and then from D meas- 
ure the line D E so that C will be in the center. The line E F 
will then be equal to the line B D. 

5. When a fence is built on a line to be surveyed, it is best to 
take an oftset either to one side or the other, and allow for it when 
the stakes are set on the true line, or the stakes may be moved back 
a distance equal to the offset as they are set. They will thus be on 
the random line, and may be corrected the same as if no offset 
had been taken. 

It is customary, after an offset has been taken, to measure back 
to the random line as soon as the obstruction is cleared, but if the 
corner be reached before this is done, the offset must not be forgot- 
ten in measuring the distance the line runs to the right or left of it. 

In doing this observe the following rules : 

(1). When the offset is taken either to the right or left and the 
offset line terminates on the opposite side of the corner, the dis- 
tance missed by the random line will be equal to the distance 
missed by the offset line, plus the offset, and it will terminate on 
the same side of the corner as the offset line. 

« B 



Fig. 31. 

In the figure an offset, A B, of 10 links was taken to the right, 
and the offset line, B C, ran 10 links to the left of the corner A. 
The random line, A B, will therefore terminate 20 links to the left 
of the corner. 

(2). When the offset line terminates on the same side as that on 
which the offset is taken. 

This involves two cases : (a) When the distance missed is 
greater than the offset, and (6) when the offset is greater than the 
distance missed. 

[a). Subtract the oftset from the distance that the oftset line 
misses the corner ; the remainder will be the distance missed by 
the random line. The termination of the random line will lie on 
the same side of the corner as the termination of the offset line. 

(b). Subtract the distance missed by the offset line from the 



MANUAL OF PLANE SURVEYING. 



101 



offset ; the difference will equal the distance missed by the random 
line. The termination of the random line will be on the opposite 
side of the corner from the termination of the offset line. 

The offset line is always parallel to the random line. 

In correcting the stakes on -the offset line, it is best to correct as 
if they were on the random line, and then move them a distance 
equal to the size of the offset, and in a direction opposite to that 
in which the offset is taken. This will put them on the true line. 

6. Sometimes, when surveys are made over hills, it is impossi- 
ble for the chain-men to see the compass or flag to which they are 
running. In this case a stake should be put up at some promi- 
nent point on the line by the surveyor, to which they may meas- 




Fig. 32. 

ure until they come in sight of the compass or flag. For instance, 
if the chain-men are down in the valley A, of the figure, a stake 
or flag should be set on the ridge, as it is impossible for them to 
see the compass at B. 

In chaining up and down hill it is frequently necessary to 
double the chain or divide it into two sections, so that it may be 
held in a horizontal position. A light steel chain is always pref- 
erable to a clumsy iron one, as it will not sag so much. 



Second Class of Obstacles. 

( 190 ). 1. Suppose the obstacle to be a large creek. The line 
is surveyed up somewhere near the edge and the flag set on the 
line on the opposite shore. In Fig. 33, let A represent the point 
to which the line is measured, and B the flag set on the opposite 
shore. From the point A, a line of indefinite length is sighted at 
right angles to A B. The compass is then set at any point not too 
near A, as C, on this line, and turned so the sights will strike B. 
The size of the angle A C B is then noted, and the point D on the 



102 



MANUAL OF PLANE SURVEYING. 



iline A E sighted at an equal angle on the other side of A C. The 
• distance from A to B will then equal the distance from A to D. 




2. From the point A, a perpendicular, A C, may be sighted and 
^another, C D, set off from its extremity. The point E, on the line 




Fig. 34. 
A C, is then found, and each of the sections, E C and E A, meas- 



MANUAL OF PLANE SURVEYING. 



103 



ured. The distance from A to B may then be found by the fol- 
lowing proportion : 

CE : EA :: CD: (x = AB); 

EAXCD 

whence A B = — — 

C hi 

3. A perpendicular is set off from the line B F at F, and another 
at A, extended to the line B D. The distances, A F, A C, and 




Fig. 35. 



D F, are measured and the distance A B, found as follows : 
(DF — AC) : AC :: AF : (x = A B) ; 

. . _ ACXAF 

whence A B - DF _ AC 

( 191 ). A great many other methods might easily be given, but 
these will suffice. 

These methods will, of course, answer equally well where the 
point B is inaccessible and at the termination of a line. 

In field-work the method that seems best adapted to the peculi- 
arities of the case should be adopted. 

Questions on Chapter XI. 

1. What is meant by an obstacle to measurement? To align- 

ment? 

2. How. many classes of obstacles are there? Name one of 

each class. 



104 MANUAL OF PLANE SURVEYING. 

3. Describe the method of spanning an obstacle by perpen- 

diculars. By an equilateral triangle. 

4. In the survey of a certain line an obstacle is met. A line 

is then surveyed 1 chain and 40 links, bearing from the 
termination of the main line so as to pass the obstacle. 
From the end of this line a perpendicular 90 links long- 
is measured back to the" original line produced through 
the obstacle. What is the distance through the obstacle ? 

5. Describe the method by symmetrical triangles, 

6. What is an offset? 

7. What is the difference between the offset line and the ran- 

dom line? 

8. An offset of 12 links is taken to the right, and the offset 

line misses 13 links to the left. How far will the random 
. line miss the corner? 

9. What two cases arise when the offset and termination of the 

offset line both lie on the same side of the corner? 

10. Give the rule for correcting the stakes on an offset line. 

11. How do you set an intermediate stake for the flag-men to 

run to when an elevation of land prevents them from see- 
ing the compass or flagstaff? 

12. When is it necessary to double the chain or divide it into 

two sections? 

13. Explain each of the methods used in the second class of 

obstacles. 

14. In Fig. 34, AE = 1.40, C E = 90, and C D = 1.10. What 

is the length of A B ? 

15. In Fig. 35, A C = 1.22, A F = 98, and D F = 1.54. What 

is the length of A B? 



CHAPTER XII. 

COMPUTATION OF AREA. 

( 192 ). In computing areas the length of all lines should be 
expressed in chains, chains and links, or links, as the case may 
be, and the areas may then be reduced to acres and decimals of 
an acre by the rules for multiplication and division of decimals. 

Thus : 

(1). 11.25 X 2.50 = 28.1250 sq ch. ; 
28.1250 -r-10 = 2.81250 acres. 

(2). 21.32 X 8 = 170.56 sq. ch. ; 
170.56 -i- 10 == 17.056 acres. 

(3). 22 X 15 = 330 sq. ch. ; 
330-^-10 = 33.0 acres. 

( 193). The following special rules are deduced from the pre- 
ceding processes : 

(1). When each dimension contains hundredths of a chain 
(links), the product maybe reduced to acres by pointing off five 
decimal places. 

(2). When one contains hundredths and the other is expressed 
in full chains, the product is reduced to acres by pointing off 
three decimal places. 

(3). When each dimension is in full chains, reduce to acres by 
pointing off one decimal place. 

If either dimension contain a fraction of a link, point off as 
many additional places in the product as are necessary to express the frac- 
tion, Thus: 

(1 ). 5.125 X 3.20 = 1.640000 acres. 

(2). 2.3725X0 = 1.18625 acres. 
(105) 



106 



MANUAL OF PLANE SURVEYING. 



1 104). 



(1). 21.52 

(2). 40.24 

(3). 16.42 

(4). 12 



EXAMPLES. 

X 12.40 = how many acres ? 
X 20.31 = " 



X18 = " 

X 13 == " 
(5). 12.165 X 15.30 = " 
(6). 15.2425X12.125= " 

( 195 ). Every tract of land is in figure a polygon, and its area 
is computed according to the rule for finding the area of the par- 
ticular polygon representing its contour. 

( 196 ). Rectangles. 

1. Multiply the length by the breadth and the product will be 
the area. 

2. If the rectangle be a square, the area maybe found by squar- 
ing one side. 

( 197 ). Parallelograms. 

Multiply the length of one side by the perpendicular distance 
between this side and the opposite. The product will equal the 
area. 

In Fig. 36 the line A B represents the perpendicular, and the 

D AC 



/ 


! 


/ 


F /5.60 




Fig. 36. 







area of the parallelogram is therefore equal to the product of 15.60 
by 7.55 = 11.778 acres. 

If either of the sides B C or D E be given instead of B E or C D, 
the perpendicular must be measured in the direction of the length 
of the figure. 



MANUAL OF PL AXE SURVEYING. 



107 



(198). Trapezoids. 

Multiply the sum of the parallel sides by the perpendicular 
distance between them, and half the product will be the area. 



D 



■ft. 90 



t3.2H 

Fig. 37. 



B C 



In Fig. 37 the line A B represents the distance between the par- 
allel sides, and the area is equal to ( (16.24 + 14.90) X 6.40) -*- 2 = 
9.4648 acres. 

When the trapezoid contains two right-angles, the line between 
them represents the perpendicular distance between the parallel 
sides. 

(199). Triangles. 

In computing the area of triangles, two general classes will be 
considered : 

1. Triangles whose base and altitude are given. 

The area of triangles of this class is found by multiplying the 
hme by the altitude and taking half the product. 




Fig. 38. 

Fig. 38 represents a triangle whose base is 18.05 and altitude 
5.90. Its area, therefore, is (18.05 X 5.90) -*- 2 = 5.32475 acres. 
Right-angle triangles belong to this class. 



108 MANUAL OF PLANE SURVEYING. 

When the three sides are given, isosceles triangles may be 
brought under this class by the following rule : 

Square one of the equal sides and subtract the square of half 
of the odd side. The square root of the difference will equal the 
altitude. 

The altitude of an equilateral triangle is found by extracting- 
the square root of the square of one side minus the square of 
half of one side. 

EXAMPLES. 

(1 ). The base is 14.75 and the altitude 2.90. What is the area of 
the triangle? 

(2). The base and perpendicular of a right-angle triangle are 5.60 
and 7.42, respectively. What is its area? 

(3). In an isosceles triangle the even sides are each 9.16 in 
length and the odd side 7.45. What is the area? 

(4). What is the area of an equilateral triangle each of whose 
sides is 7.25 in length? 

2. Triangles whose altitude is not given. 

The general rule for triangles of this class is the following : 

(a). Take half the sum of the three sides. 

(6). Subtract from the half sum each side severally. 

(c). Multiply the half sum and three remainders together. 

(d). Extract the square root of the product for the area. 




Fig. 39. 

Let Fig. 39 represents a triangle whose area is to be computed, 
Then, 10.54 + 14.60 + 8.72 == 33.86 
33.86 -4- 2 = 16.93 
16.93 — 10.54 = 6.93 
16.93 — 14.60=- 2.33 
16.93— 8.72= 8.21 



1/16.93 X 6.39 X 2.33 X 8.21 = area. 



MANUAL OF PLANE SURVEYING. 



109 



EXAMPLES. 

1. What is the area of a triangle, the sides of which are 4.20, 
2.65, 3.71, respectively? 

2. The sides of a triangle are 2.91, 6.90, and 5.42, respectively. 
What is its area? 

It is sometimes more convenient to measure the altitude, and 
avis place the triangle under the first class. 

( 200 ). . Trapeziums. 

Divide the trapezium into two triangles. The sum of their 
areas will be the area of the trapezium. 

To do this, measure a diagonal of the trapezium. 



ASJUL 




D 



/o. 7? 

Fig. 40. 



Fig. 40 represents a trapezium, one of whose diagonals has been 




Fig. 41. 



110 MANUAL OF PLANE SURVEYING. 

measured. It will be seen that its area will equal the sum of the 
areas of the triangles A B D and BCD. 

A serious mistake is sometimes made by incompetent persons by 
multiplying together the half-sums of the opposite sides for the 
area. 

When one angle of the trapezium is re-entrant, as in Fig. 41, the- 
area may be found by subtracting the area of the triangle BCD 
from that of the triangle A B D ; or it may be computed the same 
as when the angles are all salient by omitting the triangle BCD 
and measuring a diagonal from A to C. 

(201). Any Figure. 

Divide the figure into triangles and compute their areas sepa- 
rately. The sum of the areas of the triangles will be the area of 
the figure. 

The area of the tract of land represented in Fig. 42 is equal t& 




the sum of the areas of the triangles A B F, B E F, B C E, and 
CDE. 

Sometimes, when a tract of land is narrow and has one irregu- 
lar boundary, its area may be approximated by dividing it into 
trapezoids. Fig. 43 represents a tract of this kind bounded on 
one side by a creek. In cases of this kind the area of the tract is 
equal to the sum of the areas of the trapezoids that compose it. 



MANUAL OF PLANE SURVEYING. 



Ill 




Fig. 43. 

(202). 

Computation of Area by Latitudes and Departures, 

The method of latitudes and departures now to be developed is 
simple, precise, expeditious, a»d universal in application, if the 
course and distance of each of the boundary lines of the tract 
whose area is to be computed is given. 

(203). In plane surveying, meridians, like parallels of lati- 
tude, are supposed to be parallel to one another, and the latitude 
of a course is the distance between two parallels running through 
its extremities, while the departure of a course is the distance be- 
tween two meridians drawn through its extremities. In Fig. 44 
the latitude of the course A B is represented by B C, and its de- 



If 




Fig. 44. 



11: 



MANUAL OF PLANE SURVEYING. 



parture by A C. It is evident that the latitude of a course is equal 
to the difference of latitude of its extremities, and that its depar- 
ture is equal to the difference of longitude of its extremities. 

( 204). The latitudes of courses bearing north are called north 
latitudes or northings, and of those bearing south, south latitudes or 
southings. Likewise the departures of courses bearing east are 
called east departures or eastings, aud of those bearing west, west de- 
partures or westings. 

In Fig. 45 the latitudes of A B and A F are northings, and of 




A C and A D southings ; while the departures of A B and A C are 
eastings, and of A D and A F westings. 

( 205 ). North latitudes are additive and are marked with the 
sign +, plus, while south latitudes are subtraetive and marked witli 
the sign -— , minus. In the same manner, east departures are 
additive and marked +, and west departures are subtraetive and 
marked — . 

( 206 ). If we now refer to Fig. 12, we shall see that the radius 
A F, may represent the course A C, Fig. 46, whose latitude and 
departure we wish to find. Then will C E, the departure, equal 



MANUAL OF PLANE SURVEYING. 



113 



the sine of the angle B A C, and E A, the latitude, equal the cosine 
of the angle B A C. 

( 207 ). A Table of Natural* Sines and Cosines is given in the 




Fig. 46. 



Appendix, by which the latitude and departure of any course may 
be easily found. In this table the length of the sine and cosine is 
given for a radius equal to unity, for each degree and minute of 
arc between and 90°; and, hence, to find the latitude of any 
course, it is necessary only to multiply the cosine of its bearing by the 
length of the course, and to find the departure of any course, to' mul- 
tiply the sine of its bearing by the length of the course.^ 

For instance, suppose it is required to find the latitude and de- 
parture of a course bearing N 42° 33' E, and 20.22 in length. 

By referring to the Table, we find the cosine of the bearing to 
to be .73669, and the sine of the bearing to be .67623. 

Therefore, the latitude of the course will equal 
• .73669 X 20.22 = 14.8958, 
and the departure of the course will equal 

.67623 X 20.22 = 13.6733. 

In using the Table, when the bearing is 45° or less, take the de- 
grees from the top of the page and the minutes from the left-hand 

* Called natural sines and cosines to distinguish them from logarithmic sines 
and cosines. 

f In this rule observe that the angles are measured to the right and left of 
the vertical radii. . If, as in Fig. 12, they were measured from horizontal 
radii, the word "sine" would be used for "cosine," and "cosine" for 
" sine" in the rule. 



114 MANUAL OF PLANE SURVEYING. 

column, and when the bearing is greater than 45°, use the degrees 
at the bottom of the page and the minutes in the right-hand 
column. 

( 208 ). EXAMPLES. 

The course and distance is given in each of the following cases. 
Find the latitudes and departures : 

(1). N 52° 16" W, 10.12. 
This bearing is greater than 45°; so the degrees must be taken 
from the bottom of the page. Having found the double column 
marked 52°, ascend it to the line marked 16 / on the right. We 
now find the cosine to be .61199, and the sine to be .79087; there- 
fore the 



Latitude = .61199 X 10.12, 


and the 


Departure = .79087 X 10.12. 




(2). S 15° 40 7 E, 11.41. 




(3). N 21° 32' W, 19.71. 




(4). S 88° 56' E, 73.98. 




(5). N 66° 25' E, 46.12. 





(209). It will be seen that the columns in the table marked 
"sine" at the top are marked "cosine" at the bottom, and that 
those marked "cosine" at the head are marked "sine" below. 
Care must be taken to use the heading for bearings read from the 
top, i. e., for bearings riot greater than 45°; and the bottom mark- 
ings for bearings read from below, i. e., for bearings greater than 
45°. 

(210). Traverse Tables are sometimes used instead of the 
Table of Natural Sines and Cosines in determining the latitudes 
and departures of courses, and somewhat facilitate calculations in 
many cases ; but they are usually computed only to quarter- 
degrees, and it has been thought best to use in the present work 
only the more accurate method of natural sines and cosines. 

(211). In the survey of every tract of land, the sum of the 
north latitudes should equal the sum of the south latitudes, and 
the sum of the east departures should equal the sum of the west 
departures; and, hence, in plotting a survey, or making prepara- 
tions to compute the area of the tract, we have an almost infalli- 
ble means of testing the accuracy of the survey by which the 
course and distance*) f each of its boundaries were determined. 

(212). Let us now make an application to the survey of the 



MANUAL OF PLANE SURVEYING. 



115 



following described tract of land : Running N 10° E, 5.60 ; thence 
S 35° 30 / E, 4.00; thence S 55° 30' W, 4.00, to the place of begin- 
ning. 

Taking each course separately, we find the respective latitudes 
and departures. 

(1). Latitude of first course equals .98481 X 5.60 = 5.51. 
Departure equals .17366X5.60 = .97. 

(2). Latitude of second course equals .81412 X 4.00 = 3.25, 
Departure equals .58070 X 4.00 =2.32. 

(3). Latitude of third course equals .56641 X 4.00 = 2.26. 
Departure equals .82413 X 4.00 = 3.29. 

The latitude of the first course is a north latitude, and must be 
marked +, and the latitudes of the second and third courses are 
south latitudes, and take the sign — . Likewise, the departures 
of the first and second courses are east departures and should be 
marked +, while the departure of the third course is a west de- 
parture and should be marked — . 

The separate courses, with the latitude and departure for each 
one, may be entered in a diagram similar to the one used in keep- 
ing field-notes (Art. 160), and a space left at the bottom for the 
footings, as follows : 



Sta. 


Course. 


Dis. 


Lat. 


Dep. 


+ 


— 


+ 


— 


A 
B 

C 


N 10° 00' E. 
S 35° 3(y E. 
C 55° 3V W 


5.60 
4.00 
4.00 


5.51 


3.25 
2.26 


.97 

2.32 


3 29 








5.51 


5.51 


3.29 


3.29 



Fig. 47. 



( 213 ). The reason why the east departures should balance the 
west departures, and the north balance the south latitudes will be 
seen by noticing Fig. 48, which represents the above tract of land. 
The north and south lines represent the latitudes, and the east 
and west lines the departures. 

( 214 ). When the + latitudes balance the — latitudes, and the 
+ departures balance the — departures, as in the case just con- 



116 



MANUAL OF PLANE SURVEYING. 



sidered, the survey is said to " close." Usually, however, owing 
to slight inaccuracies in sighting the flag, reading the bearing of 
the line, measuring the line, or, perhaps, all combined, neither the 
latitudes nor departures balance. If the disagreement is consider- 
able, a re-survey should be made, as there is probably an error 




Fig. 48. 



somewhere in the work ; but if it is only slight, as, for instance, 1 
or 2 links in 7 or 8 chains, it is probably due to some unavoidable 
inaccuracy in the survey, and may be corrected by the following 

rule : 

Find the amount of the error for each chain, and distribute it among 
the latitudes or departures, as the case may be, in proportion to their re- 
spective lengths. Adding to those that are too small, and subtracting from 
those that are too large. 



MANUAL OF PLANE SURVEYING. 



117 



This will cause them to balance and answer all ordinary pur- 
poses. 

(215). The longitude or meridian distance of a line is its mean 
distance from an initial*line or meridian. Preparatory to-finding 
the area of a tract of land, this meridian is conceived to be drawn 
through its extreme western or eastern corner — usually the west- 
ern — and the longitude of each of the courses of the tract is com- 
puted from this meridian as a base. 

In Fig. 49 this meridian is drawn through the western corner 




Fig. 49. 



of the tract, and the lines, A B,CD,EF,G H, and M O, repre- 
sent the longitudes of the various courses. 

(216). It will be observed that there is a difference between 
longitudes and departures: The former show the mean distance 
of the line from the meridian, while the latter indicate the differ- 
ence in longitude of the two ends of the line. 

( 217 ). By referring to the figure, it will be seen that the lon- 
gitude, A B, of the first course is equal to half of its departure, 



118 MANUAL OF PLANE SURVEYING. 

a b; and also that the longitude, C D, of the second course is equal 
to c d, which equals the longitude of the first course, plus half the 
departure of the first course, plus half the departure of the second 
course, and it may easily be shown that the longitude of any course 
is equal to the longitude of the preceding course, plus half the departure of 
the preceding course, plus half the departure of the course itsef 

( 218 ). It must be borne in mind that the algebraic sum is 
meant, and that west departures, having the minus sign, are really 
subtractive. 

(219). In order to simplify the rule, and at the same time 
avoid fractions, it will be preferable to double each of the pre- 
ceding expressions and use double longitudes. The following will 
then be the general rule for finding the double longitudes of courses. 

The double longitude of the first course is equal to its departure. 

The double longitude of the second course is equal to the double longi- 
tude of the first course + the departure of the first course + the departure 
of the second course. 

The double longitude of any course is equal to the double longitude of 
the preceding course + the departure of the preceding course + the de- 
parture of the course itself. 

Computation of Area. 

(220). We are now prepared to compute areas by means of 
longitudes. Take for example the tract of land described in 
Art. 212. 

The area of the triangle ABC, Fig. 50, is equal to the area of 
the trapezoid E A B D, plus the area of the triangle BCD, minus 
the area of the triangle ACE. 

Finding the area of each of these figures, respectively, we have : 

Area of trapezoid EABD = DEX ab = the product of the 
latitude of the course A B by its longitude = (3.25 X 2.13) = .692 
acre. (See Fig. 48.) 

Area of triangle BCD = CDXef = the product of the lati- 
tude of the course B C by its longitude = (2.26 X 1.645) = .371 
acre. 

Area of triangle ACE = CEXc d = the product of the lati- 
tude of A C by its longitude = (5.51 X -485) = .267 acre. 



MANUAL OF PLANE SURVEYING. 



119 



Therefore, the area of the triangle A B C = .692+ .371— .267 
-796 acre. 

N 




Fig. 50. 

( 221 ). In computations of this kind the product of a longitude 
by a north latitude is called a north product, and by a south latitude, is 
called a south product, and the difference between the north products and 
soiUh products is the area of the tract. 

( 222 ). Hereafter double longitudes will be used, and the differ- 
ence between the north products and the south products will then 
be double the area of the tract. 

( 228 ). The different steps in the process of computation may 
be shown very nicely, and the work kept in compact form, by rul- 
ing a sheet of paper in fourteen columns, adding seven to the right 
of the seven shown in Fig. 47. In the first four of the added 
seven write the corrected latitudes and departures, in the fifth 
the double longitudes, and in the sixth and seventh the north 
and south products or areas marked 4- and — , same as latitudes. 



120 



MANUAL OF PLANE SURVEYING. 



The following will serve as an illustration, and at the same time 
indicate the process used in computation. 



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Fig. 51. 



In this example the error in latitudes and departures amounts 
to very little in each case, and might have been disregarded in 
the calculation. 



MANUAL OF PLANE SURVEYING. 121 

( 224 ). The following is the general rule for computing areas 
oy double longitudes : 

Multiply the double longitude of each course by its latitude. 

If the latitude is north or plus, write the product in the column of plus 
areas. If south or minus, write the product in the column of minus areas. 

Half the difference between the sums of the areas of these two columns 
will be the area of the tract. 

This rule holds good for any tract of land bounded by straight 
lines. 

( 225 ). When the most westerly corner of the tract can not be 
determined readily it is best to draw a plot of the tract according 
to the directions given in the chapter on Plotting. 

( 220 ). Compute the areas of the following tracts : 

:i). NSfl^E, 2.73. 
1.28. 
2.20. 
3.53. 
3.20. 

(2). S 73° 15' E,. 19.08. 

13.68. 

10.34. 

11.36. 

9.06. 

N 20° 15' E, 23.56. 
(3). South, 3.75; S 35° 00' E, 1.04; S 86° 30 / E, 5.02; N 82° 
00' E, 1.72 ; S 34° 30" E 2.46 ; S 77° 30' E, 4. 25 ; N 45° 
30' E, 9.78 ; N 2° 40" W, 233 ; West, 2.18 ; N 4° 00' E, 
1.30; N 83° 45' W, 5.35; S 76° 00' W, 1.94; S 60° 15' 
W, 2.27 ; S 76° 00' W, 3.47 ; N 73° 30" W, 2.90 ; N 57° 
30' W, 1.65 ; S 21° 00' W. 2.67. 

( 227 ). It is, of course, plain that a due north or south course 
has no departure, and that its latitude is equal to its length. 
Likewise, that a due east or west course has no latitude, and its 
departure is equal to its length. 

( 228). It is not absolutely necessary that the meridian should 
be drawn through the most westerly station in calculating the con- 
tents, but it is generally more convenient to compute from a me- 
ridian so drawn. Sometimes the surveyor imagines the meridian 
to pass through the most easterly station. 



N34° 


15 / 


E, 


N85° 


OO 7 


E, 


S 56° 


45' 


E, 


S 34° 


15' 


w, 


N56° 


30' 


w, 


S 73° 


15' 


E, 


S 19° 


30" 


W, 


N69° 


15' 


w, 


S 20° 


15' 


w, 


N68° 


00' 


W, 



122 MANUAL OF PLANE SURVEYING. 

If necessary, the areas may be expressed in the ordinary de- 
nominations of land measure (acres, roods, and rods), instead of 
in acres and decimals of an acre, by reducing the decimals to in- 
tegers. Thus, .82 of an acre = (.82 X 4) = 3.28 R. .28 X 40 == 
11.20 sq. rods. Hence, .82 acre = 3E. 11.2 rods. 

Questions on Chapter XII. 

1. Why should the length of lines be written in chains and 

links ? 

2. When links are multiplied by links, how many decimals 

are pointed off in reducing the product to acres? Links 
by chains ? Chains by chains ? 

3. Give the rule for finding the area of a rectangle. A paral- 

lelogram. A trapezoid. 

4. How is the area of a triangle found when the base and al- 

titude ?re given? When the three sides are given? 
o. How may the altitude of an isosceles triangle be found? Of 
an equilateral triangle? 

6. State the rule for finding the area of a trapezium. 

7. When may the area of a figure be computed by dividing it 

into trapezoids? 

8. What is meant by the latitude of a course? Departure? 

9. What are north latitudes? South latitudes? East de- 

partures? West departures ? 

10. Describe the Table of Natural Sines and Cosines. 

11. How do you find the latitude of a course from the Table? 

The departure? 

12. Give the rule for correcting latitudes and departures. 

13. What is meant by the longitude of a line? What is the 

difference between the longitude of a line and its depar- 
ture? 

14. State the rule for finding the double longitudes of courses. 

15. What are north products or areas? South products or 

areas ? 
1G. Give the general rule for computing areas by double longi- 
tudes. 



CHAPTER XIII. 

LAYING OUT AND DIVIDING UP LAND. 

( 229 ). No general rule can be given either for laying out or 
dividing up land, and in the present chapter only the most com- 
mon cases that arise in practice will be. considered. This is nec- 
essary in order to keep our work within its intended limits as well 
as to avoid the confusion that a multiplication of details would 
cause. * 

As a general thing, a little ingenuity on the part of the surveyor 
will enable him to devise a method to meet the exigencies of the 
case he may have on hand when it can not be reached by any of 
the rules given in this chapter. 

( 230). In the problems now to be taken up, the area and one 
or more of the boundaries are in nearly all cases supposed to be 
known, and it is required to find from these the length of certain 
other boundary lines necessary to a survey of the tract. The pro- 
cesses used in work of this kind are generally the reverse of those 
-employed in the last chapter, so that a careful study of operations 
in computing areas will materially assist in the work now before us. 

Laying Out Land. 

( 231 ). To Lay Out a Square. — The square root of the area ex- 
pressed in square chains and decimals will represent the length 
of one of its sides. 

Thus, each side of a square tract of land containing 5 acres 

equals -/50 = 7.07. 

EXAMPLES. 

1. What is the length of each side of a square tract containing 
1 acre? 

2. A piece of land in the form of a square contains 11 A. 3 R. 
26 P. What is the length of its sides ? 

(123) 



124 MANUAL OF PLANE SURVEYING. 

( 232 ). To Lay Out a Rectangle, — Divide the area by the length 
of the given side. The quotient will be the length of the required 
side. 

Thus, if a rectangle contain 4 acres, and the given length be 5 
chains, the length of the required side will equal (40 ~- d) = S 
chains. 

EXAMPLES. 

1. The area of a rectangle is 14 acres, and the length 15.00, 
what is the breadth ? 

2. The area of a rectangular tract of land equals 10 A. 2 E, 20 
P., and it is 63 rods long. What is the breadth? 

Process— 10 A. 2 K. 20 P. == 10.625 acres. 
63 rods == 15.75. 
(10.625 X 10) -*- 15.75 = 6.746. 

( 233). To Lay Out a Parallelogram. — (a). Divide the area by 
the given length. The quotient will be the perpendicular distance 
between the given sides. (Art. 197). 

( b ). Find one of the angles of the parallelogram according to 
the methods explained in Articles (22) and (23). 

(c). Divide the length of the perpendicular by the sine of the 
angle thus found, and the quotient will equal the required side. 

If the angle of the parallelogram be greater than 90°, its sup- 
plement* must be used in its stead. 

Let Fig. 52 represent the parallelogram to be laid out ; its area 




8ACBE2 



A C 12. .o o 

Fig. 52. 




I 



being 6 acres, and the length of the side A B, 12.00. Dividing 
(6 X 10) = 60 sq. chains by 12, we find the perpendicular C D, to 
be 5.00 long. 

*The sine of an angle is always equal to the sine of the supplement of the 
angle. 



MANUAL OF PLANE SURVEYING. 



125 



Suppose the angle A D E to equal 108°; then its supplement 
will be (180°— 108°) = 72°, the sine of which is .95106. 

Dividing 5.00 by .95106, we find the length of the required side, 
A D, to be 5.257. 

Had the angle BAD been used, instead of A D E, the supple- 
ment need not have been taken, as it is less than 90°. 

EXAMPLES. 

1. In a parallelogram, the area is 12 acres, the length of one 
side 14.00, and the measured angle equal to 61°. What is the 
length of the required side? 

2. A field in the form of a parallelogram is 22.00 long and con- 
tains 25 acres. The size of one of the angles is 96° 45' ; what is 
the length of the required side? 

( 234)'. To Lay Out a Right-Angled Triangle. — Let it be required 




Fig. 53. 



to lay out a right-angled triangle containing .3 acre by a line per- 
pendicular to A B, Fig. 53. 

(a). Measure the angle B A C and find its sine. 

(6). Multiply the sine by any length of base, as A D, less than 
the required base. The product will represent the altitude D E 
of the triangle A D E. 

(c). Compute the area of this triangle, and the length of the 
side A B may be found by the following proportion : 

Area A D E : Area A B C : : (A D) 2 : (A B) 2 . 

If AD= 1.20, and D E .80, the area of the triangle A D E will 
be .048 acre. Then, 

.048 : .3 : : (1.20) 2 : (A B) 2 ; whence (A B) 2 = 9, and A B = 3 chains. 



126 



MANUAL OF PLANE SURVEYING. 



The length of the side A C may be found by a similar propor- 
tion : 

Area A D E : Area A B C : : (A E) 2 : (A C) 2 . 

( 235 ). To Lay Out a Trapezoid. — Approximate the distance be- 
tween the parallel sides by treating it as a parallelogram. The 
distance thus found will be too short if the sides not parallel con- 
verge, and too long if they diverge. Let Fig. 54 represent a tract 
to be laid out or parted off. 



E.C 



A 



Fig. 54. 




B 



By dividing the area by the length of the line A B, the perpen- 
dicular is found to equal C D. The guess line E D is then meas- 
ured, and the area of the trapezoid ABDE computed. The de- 
ficiency of area is then added outside of D E, and the trapezoid 
A B F G will then contain the required amount of land. If it still 
vary a little from the exact amount, the line F G may be moved 
further out or in, as the case may be. 

In case of divergence of the sides, the overplus of area must be 
subtracted from the computed area, and the guess line moved 
back instead of further out. 




D £ 



Fig. 55. 



MANUAL OF PLANE SURVEYING. 12T 

When the difference in length of the parallel sides can be deter- 
mined without a measurement, no guess line need be surveyed, 
providing the distance between the parallel sides be known. Let 
A B C D, Fig. 55, represent a tract of land to be laid out or parted 
off the main tract by a line perpendicular to its parallel sides. 

Divide its area by the perpendicular distance between its par- 
allel sides. The quotient will be the mean length of the trapezoid. 
From this subtract half the distance D E for the shorter side, and 
add for the longer side. 

(236). To Lay Off any Figure. — When the underlying princi- 
ples of the particular problem differ from those obtaining in any 
of the cases considered, it will probably be best to depend on cor- 
rections made from guess lines, as in Art. 235, and thus reach the 
result by approximations. Yet, in many instances, easy and beau- 
tiful solutions may be reached by close observation and study. 

Dividing up Land. 

(237 ). Problems in dividing up land are such as grow out of 
division of estates, generally among heirs. This division is made 
with reference to the value of the respective shares (considering 
location, improvements, quality of soil, etc.), and not with regard 
to the quantity of land each share contains. If, however, taking 
all these things into consideration, the value of the land is uni- 
form throughout the tract to be divided, and the shares of the per- 
sons among whom it is to be partitioned are equal to each other, 
each should receive the same quantity. 

In making a partition of land no share should be taken out in 
such a way that it will injure any other share, when it possibly 
can be avoided. 

(238). The problems in dividing land introduced into this 
chapter are of the nature of those that usually come up in prac- 
tice where the land has been surveyed according to the Rectangu- 
lar System. Only simple ones have been chosen. 

( 239 ). To Divide a Rectangle into Equal Parts by Lines Parallel to 
a Side. — Divide each of the lines upon which all of these parts are 
to rest into as many equal sections as there are shares. Connect 
the extremities of these sections by perpendiculars, and these per- 
pendiculars will be the division lines of the shares. 



128 MANUAL OF PLANE SURVEYING. 

Let Fig. 56 represent the south half of a quarter-section con- 



10.00 


10.00 


10.00 


10.00 


20 
A. 


i° 


20 
A. 


20 
A. 


10.00 


10.00 - 


10.00 


10.00 



6 d f 

Fig. oQ. 

taining exactly 80 acres. The perpendiculars, a b, c d, and e /, 
divide it into four parts, each containing 20 acres. 

( 240 ). To divide a rectangle into any number of unequal parts bear- 
ing a given relation to one another, by lines running parallel to a side. — 
Suppose that, on account of the varying value of the land in the 
tract to be divided, the shares are to be to one another as the num- 
bers 1, 2, and 5. In this case, divide the base lines into parts 
bearing the same relation to one another as the shares, and con- 



5.00 | 


10.00 


25.00 


1° 1 
A. j 


20 
A. 


50 
A. 


5.00 i 


10.00 


25.00 x 



Fig. 57. 

nect the points of division by perpendiculars, as shown in the 
division of the 80 acre tract in the figure. 

Sometimes it is possible to divide land of varying quality so 
that each share shall contain its portion both of the best and 
Avorst. If we consider the unequal division in Fig. 57 to have 
been caused by the difference in quality of the land in various 
parts of the tract, it might have been possible to make the shares 
all equal by dividing the tract in the direction of its length. 



MANUAL OF PLANE SURVEYING. 129 

The figure of the shares may be almost as variable as the quan- 
tity of la nfl they contain. The rectangular form is preferred, but 
of course can not always be preserved, even in the territory sur- 
veyed according to the Kectangular System. 

( 241 ). Problems.— 1. Divide a quarter-section of land into five 
shares in the series 1, 2, 3, 4, 5, by lines running parallel to a side. 

2. The commissioners, in a certain partition of a quarter-sec- 
tion, set off the widow's dower of 30 acres in the form of a square 
in the south-east corner, and divided the remainder, by lines 
running north and south, equally among five children. What 
was the width of each share ? 

3. In a sale of a quarter-section for taxes, the lowest bid was 
for fifteen acres, and this amount was set off in the form of a square 
in the north-west corner. A few years afterward the remainder 
of the quarter was offered again, and the lowest bid was for 
twelve acres, which area was set off next to that first sold. What 
were the dimensions of each piece ? 

4. Divide the following described tract into four equal shares 
by north and south lines: South half north-east quarter, and 
east half north-east fourth north-east quarter, and south half 
north-west fourth north-east quarter. 

5. Divide a quarter-section into 7 equal shares by lines running 
north and south, and write a description of each share, giving 
metes and bounds. 

6. Divide the following tract into 5 equal shares by north and 
south lines, and write a description of each share: North half 
south-west quarter, and south-west fourth north-west quarter, and 
west half north-west fourth south-east quarter, giving metes and 
bounds. 

.7. Divide the north-west quarter, and north half north-east 
quarter, and north half north-east fourth south-west quarter, by 
east and west lines, into 3 shares that will be to each other as 1, 
. 2, and 3, and write a description of each share, giving metes and 
bounds. 

In the above examples each tract is supposed to contain exactly 
the prescribed amount of land, and the boundaries to run due 
east and west, or north and south, as the case may be. 



130 MANUAL OF PLANE SURVEYING. 



Questions on Chapter XIII. 



1. Why will a study of methods used in computation of area 

assist in laying out and dividing up land? 

2. How do you determine a side of a square from the area? Of 

a rectangle ? 

3. Give the rule for finding the required side of a parallelogram. 

4. Explain the method given for laying out a right-angle tri- 

angle. 

5. How do you lay out a trapezoid from the area and length of 

one of the parallel sides? 

6. In partitioning land, which is considered, quantity of land 

or value? 

7. Explain the method of dividing a rectangle into equal parts 

by lines running parallel to a side. 

8. In dividing lands, what figure for the shares is preferred? 



CHAPTER XIV. 



SURVEYING TOWN LOTS. 



( 242 ). The dimensions of town lots are usually given in feet, 
instead of in chains and links, and, as a general thing, the lots are 
all of th,e same size and numbered in regular order from 1 up, as 
shown in Fig. 58. The larger figures indicate the numbers of the 



*J=Z 



S 
6 



n 



sr 

22 



3 1 

H- 8 



63 



J2 



fS 
20 




S3 
2f 



WALNUT . 




S7REE 


T 




s 




/J 




S5 




'23 


to 





rv 




2B 


// 


, 30 



n 




rs 


1 


_2J__ 




3[ 


t2 




re 




38 




39 












S0« f 



Fig. 58. 



blocks, and sometimes the lots in each block are numbered sepa- 
rately ; as lot 5, block 2 ; lot 2, block 9, and so on. As the town 
grows from the original plot, the lots in each addition are fre- 

(131) 



132. 



MANUAL OF PLANE SURVEYING. 



quently numbered and referred to the particular addition to which 
they belong; as lot 8, Brown's addition; lot 7, Johnson's addi- 
tion, etc. 

( 243). The survey of the town is generally based on some in- 
dependent corner of the section in which it is situated, and im- 
portant corners in various parts of the town should be marked 
with durable monuments. Fig. 59 shows a few lots in a town lo- 
cated on a section line, and the distance is given from the section 
corner to the south-east corner of lot number 1. It will be seen 



if 



3 . 2S 



ofr' 



Fig. 59. 



& 



cO* 



that stones are placed at the north-east corner of lot number 4 and 
the south-west corner of lot number 8. These will enable the sur- 
veyor at any subsequent time to make a survey in the town with- 
out going to the section corner to find a starting point, thus saving 
him time and trouble. 

( 244 ). Lots are usually rectangular in shape and about twice 
as long as they are wide, but this is not always the case. They 
may be any reasonable shape or size that adapts them to the plan 
of the town. Likewise, streets and alleys generally cross one 
another at right angles, though by no means always. 

( 245 ). The plot of a town should always be accompanied by 
full explanations showing, 

( 1 ). The size of each of the lots. 

( 2 ). The width of each of the streets and alleys. 

( 3 ). The name of each of the additions. 

( 4 ). Any other explanations necessary to determine the bear- 
ings of any of the lines which would have to be run in a survey 
of the town. 



MANUAL OF PLANE SURVEYING. 



133 



( 246 ). Suppose that in Fig. 58 the lots are each 100 feet long, 
and 50 feet wide, the streets 50 feet wide, and the alleys 16 feet 
wide. Since the lots are rectangular, the north and south lines 
are at right angles to the east and west lines. It is evident that 
after finding a starting point, the surveyor need experience no dif- 
ficulty in the survey of any of the lots. 

If, for instance, he wishes to survey lot number 13, and can find 
no corner except the one marked with a stone at the south-east cor- 
ner of the town, he may start at the center of this stone, run west 
266 feet to the west side of Main street, and thence north 166 feet 
to the south-east corner of the lot to be surveyed. He can then 
survey the lot without any trouble. Instead of running first west 
and then north, he may run first north to the south-east corner of 
lot number 29, and thence west to the corner of the lot to be sur- 
veyed; or he may take other routes. 

(247 ). Fig. 60 represents a portion of a town in which the lots 




are of different sizes and shapes. All the lots west of Main street 
are 50 ft. wide, except number 12, which is 60 ft wide, and num- 
ber 14, which is 75 feet wide. 

Main street bears N 40° W. 



134 MANUAL OF PLANE SURVEYING. 

Lots number 15, 16, 17, 18 and 19 do not belong to the regular 
plot of the town, and are called out-lots. 

The two alleys running north into Main street, and the one be- 
tween lots 4 and 5 are each 20 ft. wide, and Main street and the 
short street between lots 2 and 3 are each 50 ft. wide. 

A stone monument marks the south-east corner of lot number 13. 

The width of each tier of lots is marked in feet at the foot of 
the tier. 

( 248 ). It is now a very easy matter to survey any of the lots 
In the regular plot of the town. Take, for example, number 11. 
To survey this lot, measure first west 280 ft., and thence north 200 
ft. to its south-west corner. From this point set off a perpendicu- 
lar and extend it to the street ; then measure north 50 ft. further 
and set off another perpendicular as before. 

(240). EXAMPLES. 

1. How would lot number 1 be surveyed ? 

2. Explain a method of surveying lot number 14. 

3. If the out-lots east of Main street were separated by lines 
perpendicular to the street, what would be the bearing of the 
lines? 

( 250 ). Town lots are measured either with a chain or tape. 

The chain used for this purpose is usually 50 or 100 feet long 
and divided into links, each 1 foot in length. It is made light, and 
as greater accuracy is generally required in surveying town lots 
than in ordinary surveying, its length should be frequently tested 
by comparison with a standard measure. The length of the chain 
Is affected by wear, temperature, and accidents. 

All measures used in surveying should be subjected to frequent 
tests. Even in surveys where* tolerable accuracy is sufficient, 
there is no excuse for neglecting anything that would be conducive 
to greater accuracy. 

Tapes used in measuring town lots usually consist of a jointed 
steel ribbon, but sometimes a linen tape through which a fine 
brass wire is interwoven with the thread, is used. Common linen 
tapes contract when wet and are not trustworthy. The steel tape 
is the best. 



MANUAL OF PLANE SURVEYING. 135 

Questions on Chapter XIV. 

, 1. How are town lots numbered? 

2. Upon what is the survey of a town usually based? 

3. What advantage is there in having important corners marked 

by monuments? 

4. What is the usual shape of town lots ? 

5. State the explanations that should accompany the plot of a 

town. 

6. A street, bears N 29° 32' E. What is the bearing (obverse 

and reverse) of a line perpendicular to it? 

7. What kind of measures are employed in the survey of town 

lots? 

8. How is the length of the chain affected ? 



CHAPTER XV. 

PLOTTING. 

(251). Plotting is the operation of drawing to a scale upon 
paper the lines of a survey, so that the plot will be a correct rep- 
resentation of the actual lines surveyed. 

( 252 ). The instruments used in plotting are a drawing board , 
t-square, ruler, drawing pen, dividers, protractor, and a diagonal 
scale. 

1. The drawing board should be made of pine, and its surface 
should be perfectly smooth and level. The paper is fastened to 
the drawing board while the plot is drawing. Perhaps the most 
suitable size for a drawing board is about 30 inches square, 
but 24 by 28 inches makes a very nice board. The paper should 
be stretched evenly^ and the edges pulled down over the edges of 
the board and glued or tacked. 

2. The t-square, so called on account of its resemblance to the 
letter T, consists of a thin blade with parallel edges, to which is 
attached a cross-head somewhat thicker than the blade, so as to 
form a shoulder. The blade is usually about 24 inches in length, 
and the cross-head about 10 inches. By laying the blade on the 
paper and pressing the shoulder against the edge of the drawing 
board, as shown in Fig. 61, perpendiculars may be drawn to any 
edge of the paper. 

3. A good box-wood ruler, about 12 inches long, divided to 16ths 
of an inch, will answer every purpose in plotting. This ruler 
should have one beveled edge, upon which the divisions are 
marked, and one projecting edge, along which the pen should be 
pressed in drawing lines. 

4. The drawing pen consists of two steel blades, whose distance 
apart is regulated by a thumb-screw. A little practice will en- 

(136) 



MANUAL OF PLANE SURVEYING. 



187 



able any person to draw nice smooth lines of any desirable width 
with the drawing pen. India ink should be used, as it flows more 
slhoothly from the pen than common ink. 




Fig. 61. 



5. The dividers or compasses is an instrument used in drawing 
arcs, sub-tending angles, etc., and consists of two arms which open 
and shut by a hinge joint at the end. Each of these arms termi- 
nates in a sharp point, and one, if not both, is usually jointed so 
as to permit the point to be taken out and a drawing pen put in 
its stead. In drawing large arcs a lengthening bar is inserted in 




Fig. 62. 

the jointed arm. Fig. 62 represents a pair of plain dividers, It 
will be seen that the arms are not jointed in this pair. 

6. The protractor is an instrument used in laying out angles. 
It is usually nothing more than a semicircle divided to degrees, 
half-degrees, or quarter-degrees. The degrees are numbered from 
0° to 180° in one or both directions from opposite extremities of 
the arc. The best protractors are made of silver or German silver, 
but the more common ones are made of brass or horn, and some- 
times of paper. Fig. 63 represents a small protractor. 



138 



MANUAL OF PLANE SURVEYING. 



Where great accuracy is required, protractors are supplied with 
an arm to which a vernier, like the compass vernier, is attached. 

Sometimes rectangular protractors are used instead of semi- 
circular. 

7. The diagonal scale of equal parts is a flat scale a given num- 




Fig. 63. 

ber of units, say inches, in length, and has the space devoted to 
one unit at the end divided by diagonals as shown in Fig. 64. 
These diagonals with the assistance of the lines running parallel 
to the edges of the scale, enable a person to take the length of a 

U18.SAS.2J 



LED D / / t-Qi 



ti n i \ \ \ rVM 



iiliiiiii 



llltllllli 



ii / i 



it ri- 



ff i / 



I f 



/ \E\ \ I 



11111 



III// 



in ri 



LLLLLLLL 



I l l l i 



l I 



03 



W 



05 



Od 



07 



08 



.09 



Fig. 04. 



line to j^q of the unit of the scale. If the unit of the scale be 

1 inch, then the length of a line may be taken to t J-q °f an inch. 
( 253 ). In drawing plots and maps a unit of the scale repre- 
sents a certain number of units of the line to be represented on the 
plot. Suppose the real line is 20 chains long, and it is to be plot- 
ted to a scale of 5 chains to the inch. The line on the plot will 
therefore be (.20-7-5) — 4 inches long. Fig. 65 represents a line 

2 chains in length plotted to different scales. 



MANUAL OF PLANE SURVEYING. 139 



1 in. = 5.00. 



1 in. = 2.00. 



1 in. = 1.00. 
Fig. 65. 

( 254 ). Let us now employ the drawing instruments in plot- 
ting lines. 

Suppose an east and west line 2.27 in length is to be drawn to 
-a scale of 1 chain to an inch. 

Arrange the drawing paper with its edges parallel to the edges 
of the board, and then place the t-square, as shown in Fig. 61, 
with its shoulder fitting squarely to the left-hand edge, and the 
edge of the blade just moved up to the point from which the line 
is to be drawn. Then spread the dividers so that when one arm 
is placed two units from the inner edge of the divided unit and 
on the line marked .07, the other will just reach the point where 
this line crosses the line marked .2 at the top of the scale. The 
arms then embrace the proper length of line. 

Next place one arm of the dividers against the t-square with its 
point on the point from which the line is to be drawn and swinging 
the free arm round in the proper direction until it too touches the 
same edge of the blade. Connect these two points by a line with 
the drawing pen, and it will be the required line. 

In like manner, a line 1.25 long- to the same scale may be em- 
braced by the dividers by placing one point at a on the diagonal 
scale, and the other at e; and a line 1.40 long by placing one 
point at c and the other at the point marked .4 at the top of the scale. 

If the diagonal scale is not long enough to permit the required 
line to be taken off, it may be extended by means of a ruler. 

(255). EXAMPLES. 

Draw lines representing the following distances : 



(1). 


2.50; 


scale — 


lin. 


= 1.00. 


(2). 


3.79; 


scale — 


-1 in. 


= 1.00. 


(3). 


4.75 ; 


scale — 


-1 in. 


= 2.00. 


(4). 


6.42; 


scale — 


-1 in. 


= 5.00. 


(5). 


10.00 ; 


scale — 


-1 in. 


= 5.00. 


(6). 


12.31 ; 


scale — 


- 1 in. 


= 10.00. 



140 MANUAL OF PLANE SURVEYING. 

(256). Rectangular tracts of land may be plotted with the in- 
struments used in drawing lines already described. 

Take, for instance, a rectangular tract 7.15 long, 4.35 wide. 

First draw a line representing the length of the tract, then 
another perpendicular to this at one end representing the breadth, 
then from the end of this another parallel to the first and of equal 
length, and close by connecting the extremities of the first and 
third with one another. 

If no diagonal scale is at hand, a common ruler will answer for 
rough work. 

( 257 ). The protractor is used in nearly all cases where the 
courses and distances of the boundaries of the tract to be plotted 
are given, and its use will now be explained. 

The bearing of a line represents the angle the line makes 
either with the magnetic meridian or the true meridian drawn 
through the point from which the bearing is taken ; and to deter- 
mine this angle and represent the line on the plot, meridians 
should be drawn through each station of tne survey as soon as the 
station is located. 

Begin at any important station to draw the plot by laying out a 
meridian on the proper part of the paper and locating the station 
on this meridian; then draw the first course at the proper angle 
with this* meridian, producing it the required distance, as explained 
in Art. 254; then draw another meridian through the other ex- 
tremity of the course, and lay out the second course in the same 
way ; proceed in this way until the lines are all drawn, and the last 
line should terminate at the station taken as the starting point in 
the plot. 

( 258 ). Let us now draw a plot of the following tract of land ; 
N •62° 45 / E, 9.25 ; thence S 36° E, 7.60 ; thence S 45° 30' W, 10.40; 
thence N 31° 30' W, 10.00. 

In this case we may commence at the first station in the de- 
scription. Draw a meridian as N S, Fig. 66 and locate the sta- 
tion at some point, as A, on the meridian. Then place the pro- 
tractor so its center will fall on the station and its edge coincide 
with the meridian, and with the point of a pin mark the termina- 
tion of an arc of 66° 45' from the north end of the protractor. 
Then draw the line A B from the station through this point and 
determine its length by the method explained in Art. 254. Draw 



MANUAL OF PLANE SURVEYING. 



141 



B C, C D, and D A in the same manner, and the plot will be com- 
plete. 

The plot, Fig. 66> is constructed to a scale of 5 chains to the 
inch. 

In northerly courses the angle or bearing should be read from 
the north end of the protractor, and in southerly courses from the 
south end. 




Fig. 66. 



If the last course lack but a little of terminating at the first 
station, the discrepancy may be the result of the imperfection of 
the instruments employed ; but if the extremities of the lines are 
a considerable distance apart, it is probable that a mistake has 
been made somewhere. If it be tested by latitudes and departures, 
and they balance (Art. 212), the mistake is in the plot, but if they 
do not balance, the error is in some of the previous work. 



142 



MANUAL OF PLANE SURVEYING. 



The descriptions given in Art. 226 may be used as examples in 
plotting. 

Various other methods are also used in drawing plots, but the 
one given is, perhaps, the most speedy and simple, and will 
answer every purpose. 

. (259). The pantograph is an instrument used for copying 
plots, etc., either in a reduced or enlarged form. It consists of 
four rulers arranged somewhat in the form of a parallelogram. By 
fastening the instrument on the drawing-board and moving a 
point on one arm along the plot to be copied, another arm to 
which a pencil is attached sketches a precise copy on the sheet 
placed under it. 

(260). Buildings, springs, etc., maybe located on the plot if 
their courses from certain points on the boundaries of the tract 
are known. 

For example, in surveying the west half of a quarter-section, a 
line from the north-east corner to the north-west corner of a house 
was found to bear S 45° W, and one from a point 12.00 west of the 
north-east corner was found to bear S 13° E. While constructing 
the plot these lines may be laid out from the proper places with 
the protractor, and the place where they meet will be the north- 
west corner of the house, as shown in Fig. 67. 




Fig. 67 



MANUAL OF PLANE SURVEYING. 143: 

( 261 ). Plots and maps may be colored with crayon pencils or 
with water-colors; but when water-colors are used care must be 
taken to keep them from running into one another and injuring 
the shades. The paper should be dampened preparatory to ap- 
plying them. For inexperienced persons, crayons will prove the 
most satisfactory. 

Questions on Chapter XV. 

1. What is plotting? 

2. Name the instruments used in plotting. 

3. Describe the diagonal scale. 

4. Describe the method of using the diagonal scale. 

5. Give the length of each of the following lines plotted to 

scales of 1 chain to an inch, 2 chains to an inch, and 10 
chains to an inch : 12.50; 15.00; 18.375; 11.25. 

6. Describe the method of using the protractor. 

7. Why should the last course in a plot terminate at the first 

station ? 

8. For what is the pantograph used ? 

9. How may buildings and other objects be located on a plot? 
10. How are plots and maps colored? 



CHAPTER XVI. 

SURVEYING WITHOUT A COMPASS. 

(262). A great many surveys can be made without the com- 
pass, and a few pages will now be devoted to the consideration of 
the most common cases in which it may be dispensed with. It 
must be borne in mind, however, that the compass could be ad- 
vantageously used in nearly all the cases here cited, and that the 
methods given are intended fpr use only in emergencies. 

(263). Setting Corners. — Where two witness trees taken to a 
corner can be found, the corner may be located from them by 
their distances measured from the sides upon which the blazes are 
made. Suppose one tree is 15 links from the corner, and the other 
19 links. Measure off 15 links on one end of a cord and 19 links 
on the other end, and tie a knot where the two measurements ter- 
minate. Then have the long end of the cord held against the 



W/TNE5S 



*%^ WITNESS. 

Fig. 68. 

blaze on the most distant tree, and the short end against the blaze 
on the other. Stretch both ends of the cord tightly, and the knot 
will mark the corner, as shown in Fig. 68. 

The corner is always in front of the blazes on the trees. 

The distances of the witness trees from the corner may be found 
from the field-notes of the tract. 

(144) 




MANUAL OF PLANE SURVEYING. 145 

Where only one witness can be found, the corner can not be 
located with certainty without a compass. 

This same method may be employed, slightly modified, in lo- 
cating a corner by the two lines meeting there, when the length of 
each line and the location of each of the corners at the other end 
of each, are known. 

(264). Establishing Lines. — When one corner is visible from 
the corner at the other end of the line, a stake may be put up, and 
intermediate points on the line may be marked at pleasure. 

When one corner is not visible from the other, but its direction is 
approximately known, the line may be " ranged '' from one to the 
other. To do this, put up a stake or flag at the corner from which 
the line is ranged, and at a certain distance, say 50 or 100 steps, 
in the direction of the other corner, set up another stake or flag. 
Then walk ahead an equal distance and set another stake in line 
with the first and second. Proceed in the same manner, always 
setting the stakes at equal distances from one another, and ranging 
the last one with the two previously put up, until the other corner 
is reached. The distance that the line misses, either to the right 
or left, can then be noted, and the stakes corrected in a manner 
entirely similar to that already explained. 

For instance, if there are 12 stakes on the line, and it termi- 
nates 30 links to the right of the corner, each stake must be moved 
to the left. The distance it is to be moved is found by dividing 
the distance missed by the number of stakes and multiplying the 
quotient by the number of the stake from the starting point. In 

f 11 X 301 
this case, the 11th stake must be moved to the left — — — =27 J 

( 10 V 301 I 12 J 

links, the 10th — — — = 25 links, and so on. 

The stake put down at the corner at starting is not counted, and 
the next is called the first. 

Of course, the location of the corners must be known before the 
line can be established. 

( 265 ). Setting Out Perpendiculars. — Almost any kind of a con- 
trivance with two lines of sight at right angles to one another, 
will answer for this purpose. It may be a sort of cross-staff with 
four upright sights provided with slits or threads, two marking- 
each line of sight. The sights need not be more than 18 inches 
10 



146 MANUAL OF PLANE SURVEYING. 

apart, and the apparatus should be made to rest on a staff about 
4 J feet high. It may be rude in construction, but the lines of 
sight should be exactly at right angles to one another. 

( 266 ). Eectangular tracts of land may be readily surveyed in 
many instances with this instrument, but it should be used with 
care in independent divisions of the section, as they are not often 
exactly rectangular in form,. One of the lines of sight may also 
be used in sighting lines, and is a good substitute for the method 
of " ranging " described in the preceding article. 

(267). Measurements. — Lines may be measured with a cord, 
tape -line, or pole, and distances may be given .in feet or links, a& 
best suit the case at hand. 



APPENDIX. 



ABSTRACT OF DECISIONS. 



(147) 



ABSTRACT OF DECISIONS 



OF THE 



UNITED STATES AND VARIOUS STATE COURTS 



RELATING TO CONTRACTS, SURVEYS, ETC. 



(Nearly all of the following Decisions have been taken by per- 
mission from Dunn's Land Decisions, a valuable book for sur- 
veyors, published by George H. Frost, New York.) 

Boundaries. 

1. Course and distance must yield to natural and artificial ob- 
jects of description. Gaveny vs. Hinton, 2 Greene (Iowa) 344. 

2. Boundaries marked on the land are to govern courses and ! 
distances. Blaisdell vs. Bissell, 6 Barr (Pa.) 478. 

3. The lines marked on the ground constitute the actual sur- 
vey and control the return of the surveyor, even where a natural 
or other fixed boundary is called for by the survey, though the 
space between the two is but tw r elve perches in breadth. Walker 
vs. Smith, 2 Barr (Pa.) 43; Hall vs. Tanner, 4 Barr (Pa.) 244. 

4. A grant called for a certain number of poles " to a stake 7 
crossing the river." Held, that the line must cross the river, 
though the distance terminated before entering it. Whiteside ys„ 
Singleton, 1 Meigs (Tenn.) 207. 

5. A survey must be closed in some way or other. If this can 
be done only by following the course the proper distance, then it 
would seem that distance should prevail ; but when distance falls 

(149) 



150 APPENDIX, 

short of closing, and the course will do it, the reason for observing 
distance fails. Doe vs. King, 3 How. (Miss) 125. 

6. Where a deed describes lands by its admeasurements, and, 
at the same time, by known and visible monuments, these latter 
shall govern. May hew vs. Norton, 17 Peck (Mass.) 357; Massen- 
gille vs. Boyles, 4 Humph. (Tenn.) 205; Woods vs. Kennedy, 5 
Monr. (Ky.) 174; Nelson 'w. Hall, 1 McLean (U. S.) 518; Camp- 
bell vs. Clark, 8 Mis. 553. 

7. The rule that monuments control in boundaries is, however, 
not inflexible ; and in case where no mistake could reasonably be 
supposed in the courses and distances, the reasons of the rule were 
held to fail, and the rule itself was not applied. Davis vs. Bains- 
ford, 17 Mass. 207. 

8. A line is to be extended to reach a boundary in the direc- 
tion called for, disregarding the distance. Witherspoon vs. Blanks, 
1 Taylor (N. C.) 110. 

9. If a vendor hold two tracts adjoining, and sell a certain 
quantity by metes and bounds, though the deeds call for one tract, 
yet if the metes and bounds run into the other, the purchaser 
shall hold according to the metes and bounds. Wallace vs. Max- 
well, 1 J. J. Marsh (Ky.) 447; Mundell vs. Perry, 2 Gill & Johns 
<Md.) 206. 

10. Posts set up at corners, between adjoining owners of land, 
•control the calls for course and distance and establish the bound- 
ary where they are mentioned and recognized in the deeds. Al- 
shire vs. Hulse, 5 Ham. (Ohio) 534. 

11. Where land is described as running a certain distance by 
admeasurement, to an ascertained line, though without a visible 
boundary, such line will control the admeasurement and de- 
termine the extent of the grant. Flagg vs. Thurston, 13 Pick. 
(N. Y.) 145 ; Carroll vs. Norwood, 5 Har. & J. (Md.) 163. 

12. Where the line or course of an adjoining tract, being suffi- 
ciently established, are called up in a patent or deed, the lines 
shall be extended to them without regard to distance. Cherry vs. 
;Slade, 3 Murph. (N. C.) 82. 

13. Where the boundaries of land are fixed, known, and un- 
questionable monuments, although neither courses, nor distances, 
nor the computed contents correspond, the monuments must gov- 
ern. Pernam vs. Wead, 6 Mass. 131 ; Calhoun vs. Wall, 2 Har. & 
McHen. (Mo.) 416. 



APPENDIX. 151 

14. If a deed from the government of the U. S., or an individ- 
ual, describes land as partly bounded by a river, the river bound- 
ary will be adhered to, though it does not correspond with estab- 
lished corners and monuments. Shelton vs. Mauphin, 16 Mo. 124. 

15. If nothing exists to control the call for courses and dis- 
tances, the land must be bounded by the course and distances of 
the grant, according to the magnetic meridian ; but courses and 
distances must yield to natural objects. 16 Ga. 141. 

17. The corners established by the original surveyors of public 
lands under the authority of the United States, are conclusive as 
to the boundaries of sections and divisions thereof, and no error 
in placing them can be corrected by any survey made by individ- 
uals or by a state surveyor. Arnier vs. Wallace, 28 Miss. 556. 

18. Whenever natural or permanent objects are embraced in 
the calls of either a survey or a patent, these have absolute con- 
trol, and both course and distance must yield to them. Brown vs. 
Huger, 21 Howard (U. S.) 305. 

19. In determining boundaries under a grant, natural objects, 
as landmarks, are to be considered before courses and distances. 
Daggett vs. Wiley, 6 Fa. 482. 

20. Where adjoining proprietors abut on opposite banks of a 
stream, their boundary line will follow the natural and imper- 
ceptible alterations in its course, but not changes caused by arti- 
ficial means. Halsey vs. McCormick; 3 Kernan (N. Y.) 296. 

21. When lands are described in a deed or grant as bounded by 
river not navigable, the center of the stream is to be considered 
the boundary. Claremont vs. Carlton, 2 N. Hamp. 369 ; Palmer 
■vs. Mulligan, 3 Caines (N. Y.) 407, 319 ; Hayes vs. Bowman, 1 
Hand (Va.) 417; Ingraham vs. Wilkinson, 4 Pick. (Mass.) 268; 
Oavil vs. Chambers, 3 Ham. (Ohio) 496 ; Brown vs. Kennedy, 5 
Har. & J. (Md.) 195; Arnold vs. Mundy, 1 Halst. (N. J.) 1. 

Quantity of Land. 

22. A conveyance by metes and bounds will carry all the land 
contained in them. Belden vs. Seymour, 8 Conn. 19; Jackson vs. 
Ives, 9 Cow. (N. Y.) 661. Although it be more or less than is 
stated in the deed. Butler vs. Widger, 7 Cow. (N. Y.) 723. 

23. Where a specified tract of land is sold for a gross sum, the 
boundaries of the tract control the description of the quantity it 
contains, and neither party can have a remedy against the other 



152 APPENDIX. 

for an excess or deficiency i#the quantity, unless such excess or 
deficiency is so great as to furnish evidence of fraud or misrepre- 
sentation. Voorhees vs. De Meyer, 2 Bar. Sup. Ct. Rep. (N. Y.) 87, 
24. Where a person purchases land by metes and bounds said 
to contain a certain number of acres, more or less, he is entitled 
to all the land within the limits, whatever the number of acres 
may be. Bratton is. Clawson, 3 Strobh. (S. C.) 127. 

24. Quantity, although the least reliable and last to be resorted 
to of all descriptions in a deed, in determining the boundaries of 
the premises conveyed, may sometimes be considered in corrobora- 
tion of other proof. McClintock vs. Rogers, 11 Ills. 279. 

Figure oe Tracts of Land. 

25. If the order for a survey of land do not certainly determine 
the form in which it should be made, the survey ought to be in a 
square. Kennedy vs. Paine, Hardin, 10. 

26. "Seventy acres, being and lying in the south-west corner ' r 
of a section, is a good description, and the land will be in a 
square. 2 Ham. (Ohio) 327; Cockrell vs. McQ.ninn, 4 Monr. 
(Ky.) 63. 

27. The rectangular figure will be preserved in preference to 
any other in fixing locations. Massie vs. Watts, 6 Cranch, 148 ; 
Holmes vs. Trout, 7 Pet. 171. 

Acquiescence in Boundaries. 

28. Acquiescence for a long time (e. g. for eighteen years), in 
an erroneous location, is conclusive on the party making or acqui- 
escing in such location. Rockwell vs. Adams, 6 Wend. (N. Y.) 469. 

29. Where a boundary is disputed between parties who own ad- 
joining tracts, and the parties employ a surveyor, who runs out 
the line, and marks it on a plat in their presence, as a boundary, 
after twenty years corresponding possession, they are concluded 
by it. Boyd vs. Graves, 4 Wheat. 513. 

30. An acquiescence for twenty years is, as a general rule, nec- 
essary to support an implied agreement in respect to a boundary 
different from that clearly expressed in the title deeds. Hall vs. 
Cox, 7 Ind. 453. 

31. Where two persons own equal parts of a lot of land in sev- 
eralty, but not divided by visible monuments, if both are in pos- 
session of their respective parts for fifteen years, acquiescing in an 



APPENDIX. 153 

imaginary line of division during that time, that line is thereby 
established as a divisional line. Beecher vs. Parmele, 9 Ver. 352 ; 
also 18 Ver. 395. 

32. Maintaining a fence for many years is strong but not con- 
clusive evidence of limitation of claim to the boundary. Potts vs. 
Everhart, 26 Pa. 493. 

33. A party is precluded upon principles of public policy, from 
setting up or insisting upon a boundary line, in opposition to one 
which has been steadily adhered to upon both sides for more than 
forty years. Baldwin vs. Brown, 16 N. Y. 359. 

34. A division fence of more than twenty-one years' standing, 
although crooked, constitutes the line between adjacent land own- 
ers, even though the deeds of both parties call for a straight line 
between acknowledged landmarks. McCoy vs. Hance, 28 Tenn. 149. 

35. Where adjoining proprietors, being unable to ascertain the 
division line, agree verbally upon a certain line, the agreement is 
binding, and improvements by one up to the line is notice thereof 
to a purchaser from the other. Houston vs. Sneed, 15 Texas, 307. 

36. An ancient line of division marked on the ground by ad- 
joining owners, and afterwards acted upon by them, will become 
the boundary between the lots, although different from the line 
described in the original deeds. Hathaway vs. Evans, 108 Mass. 267. 

37. A possession for twenty years of a part of the land in dis- 
pute, in reference to a line conflicting with another tract, of which 
another party may be also in actual possession, but outside of the 
disputed territory, may be enough to presume the execution of a 
deed conveying the land in dispute to the party in possession. 
Amick vs. Holman, 13 Shobh. (S. C.) 132. 

38. Parties are not bound by a consent to boundaries, which 
have been fixed under an evident error, unless perhaps by the pre- 
scription of thirty years. Gray vs. Couvillon, 12 La. Ann. 730. 



TABLE OF 

NATURAL SINES AND COSINES. 



(155) 



156 



TABLE OF NATURAL SINES AND COSINES. 



5 

( o 

i 

2 
3 
4 
5 
6 
7 



s 10 

<! ii 

; 12 
13 

14 

* 15 

<J 16 

< IT 

<; 18 

; 19 

< 20 
) 21 
5 22 

23 

) 24 

< 25 
( 26 

< 27 
( 28 
( 29 

< 30 

31 

( 32 

( 33 

( 34 

( 35 

'; 36 

< 37 



38 

) 39 

) 40 

? 41 

) 42 

; 43 

I 44 

( 45 

> 46 

( 47 

< 48 

< 49 

< 50 



0° f 


1 


o 


Sine. Co in. 


Sine. 


Cosin. 

.99985 


00000 


One. 


.01745 


00029 


One. 


.01774 


.99984 


.00058 


One. 


.01803 


.99084 


.00087 


One. 


.01832 


99983 


.00116 


One. 


.01862 


.99983 


.0SH45 


One. 


.01891 


.99982 


.00175 


One. 


.01920 


.99982 


.00204 


One. 


01949 


.99981 


.00233 


One. 


.01978 


.99980 


.00202 


One. 


.02007 


.99980 


.00291 


One. 


.02030 


99979 


.00320 


.99999 


.02065 


.99979 


.00349 


.99999 


.0209 1 


.99978 


.00378 


.99999 


.02123 


.99977 


.01407 


.99999 


.02152 


.99977 


.00436 


.99999 


.02181 


.99976 


.00465 


.99999 


.02211 


.99976 


.00495 


.99999 


.0224i» 


.99975 


.00524 


.99999 


.02269 


.99974 


.00553 


.99998 


.02298 


.99974 


.00582 


.99998 


.02321 


.99973 


.00611 


.99998 


.02350 


.99972 


.00610 


.99998 


.02385 


.99972 


.00009 


.99993 


.02414 


.99971 


.00698 


.99998 


.02443 


.99970 


.00727 


.99997 


.0247; 


.99969 


.00756 


.99997 


.02501 


.99969 


.00785 


.99997 


.02530 


.99968 


.00814 


99997 


.02500 


.99967 


.00S44 


.99996 


.02589 


.99966 


.00873 


.99996 


.02618 


.99966 


.00902 


.99996 


.02617 


.99965 


.00931 


.99996 


.02676 


.999(54 


.00900 


.99995 


.02705 


.99963 


.00989 


.99995 


.02734 


.99963 


.01018 


99995 


.02763 


.99962 


.01047 


.99995 


.02792 


.99961 


.01076 


.99994 


.02821 


.99960 


.01105 


.99994 


.02859 


.99959 


.01341 


99994 


.02879 


.99959; 


.01164 


99993 


.02908 


.99958| 


.01193 


.99993 


.02938 


.999571 


.01222 


.99993 


.02967 


.99956! 


.01251 


.99992 


.02996 


.99955* 


.01280 


.99992 


.03025 


.99954 


.01309 


.99991 


.03054 


.99953 


.01338 


.99991 


.03083 


.99952. 


.01367 


.99991 


.03112 


.99952 


.01396 


.99990 


.03141 


.99951! 


.01425 


.99990 


.03170 


.99950; 


.01454 


.99989 


.03199 


.999491 


.01483 


.99989 


.03228 


.99948! 


.01513 


.99989 


.03257 


.99947! 


01542 


.99988 


.03280 


.99946 


.01571 


.99988 


.03316 


.99945 


.01600 


.99987 


.03345 


.99944 


.01629 


.99987 


.03374 


.99943' 


.01658 


.99986 


.03403 


. 99942 i 


.01687 


.99986 


.03432 


.99941 j 


.01716 


.99985 


.03461 


.99940 


.01745 


.99985 


.03490 


1.99939; 


Cosin. | Sine. 


Cosin. | Sine. | 


8 


9° 


8 


8° 



Cosin. ' 

Mm.) ! 

.99938! 
.99937! 
.99936! 
.99935: 
.99934 
.99933 
.99932 
,99931 
.99930 
.99929 
.99927 
.99926 
.99925 
.99924 
.99923 

.99922 

.99921 

.99919 

.99918 

.99917 

.99916 

. 99915 

.99913 

.99912 

.99911 

.99910(1 

.99909!' 

.99907! ! 

. 999C6 1 ; 

. 999051 ; 

.99904! 

.9! -902 I 

.99901 

.99900 

.99898 

.99897 



,99894 
99893 

99892 
99890 
99889 



.99886 
.99885 

.99883 

.99882 



.99879 
.99878 
.99876 
.99875 
.99873 
.99872 
.99870 
.99869 
.99867! 
.99866 
.99864 
.99863 



Sire 

05234 

05263 

05292 

05321 

05350 

05319 

05408 

,05437 

,05466 

.05495 

.05524 

,055r3 

.05582 

.05611 

.05640 

.05669 

.05698 
.0572' 
.057C6 
.057F5 
X5814 
.05844 
.05873 
.C59l 2 
.05931 
.05(60 
.05969 

.ceo -8 

.(0047 
.C6G76 
.16105 

.06131 
.C6162 

.06192 
.00221 
.06250 

.00279 
.C6::08 
.06337 
.06306 
.06395 
.06424 
.06153 
.00482 
.06511 
.06540 

.06569 
.C6598 
.00627 
.06656 
.06085 
.06714 
.06743 
.00773 
.00802 
i. 06831 
.06860 
.06889 
1.06918 
.06947 
.06976 



Cosin. Sine. 



99863 .06976 

99861 !|. 07005 
,998601 .07034 
9: 858! .07063 
99857 .01092 

,99855! .01121 



.99854 
.99852! 
.99851 l 
.09849 
.P9847 
.99846 
.99844 
,99842 
.99841 
.99839 

.99838 
.90886 
.99834 
.99633 
.99831 
.S98i9 
.99827 
.99826 



.99822 
.99821 
.99819 
.9981' 
.99815 
.99813 

.99812 

.99810 
.99808 
.99806 
.99804 



Cosin. I Sine. 

87° 



.99799 
.99197 
.99195 
.99793 
.99791 
.99790 
.99788 
.99186 

.9978^ 
.99182 
.99180 
.99778 
.99776 
.99774 
.99772 
.99170 
.99768 
.99766 
.99764 
.99762 
.99760 
.99758 
.99756 



Cosin. j £ine. 



86° 



.071501 
.07119 

1 .07208: 
! .01237! 

;.C7ioo 

1.01295 
.07324 

j. 01353 
! . 01882 
|. 07411 

.07440 
.07409 

.07498 
.07527 
.07556 
.07585 
.07614 
.01643 
.01672 
.01101 
.07130 
.01159 
.077*8 
l . 01817 
i. 07846 

1.07*75 
j -07904 
1.07983 
.07962 
.07991 
1.08020 
.08049 
.08078 
.08107 
.08136 
.08165 
.08194 
.08223 
.06252 
A 8281 

.08310 
.06339 
.08368 
.08397 
.08426 
.08455 
.08484 
.08513 
.08542 
.06571 
.08600 
.(8629 
.08658 
.08687 
.08716 



Cosin. 1 

^99750 
.99154 
.99752 
.99750 
.99748 
.99746 
.99144 
.99742 
.99740 
.99188 
.99136 
.99734 
.99731 
.99729 
.99727 
.99725 

.99723 

.99721 
.99719 
.99716 
.99714 
.99712 
.99710 
.99708 
.99705 
.99703 
.99701 
.99699 
.99096 
.99694 
.99692 

.99689 

.99087 

.99085 

.99i:83 

.99680 

.«. 96' 

.996' 

.99673 

.99671 

.99668 

.99666 

.99664 

.99661 

.99659 

.99657 

.99(54 
.99652 
.99649 
.9964' 
.99644 
.99(542 
.99639 
.99637 
.99635 
.99632 
.99630 
.99627 
.99625 
.99622 
.99619 



85° 



M. ■•; 

60 
59 I 

58 > 
57 ' 
56 I 
55 

54 < 
53 ) 
52 

51 < 
50 
49 

48 ) 
47 ■ 
46 
45 

44 ) 

43 ' 
42 
41 
40 ( 



£7 
36 
35 

34 ; 

83 
32 
31 
30 

29 

88 i 

27 ) 

26 ; 

25 ) 
24 

23 ( 
22 < 
21 I 
20 

19 ' 
18 S 
17 
16 

16 > 
14 ( 
13 
12 

11 

10 > 

2 

! 

6 \ 

1 

8' 



TABLE OF NATURAL SINES AND COSINES. 



157 




158 



TABLE OF NATURAL SINES AND COSINES. 





10° 


M. 


Sine. 


C> in. | 

.98481 





.1736.-) 


1 


.17393 


.98476 


2 


.17422 


.98471! 


3 


.17451 


.98466 s 


4 


.17479 


.984611 


5 


.17508 


.9345) 


6 


.17537 


.93450 


7 


.17565 


.98445 


8 


.17594 


.93440; 


9 


.17623 


.98435; 


10 


.17651 


.98430 


11 


.17680 


.98425; 


12 


.17703 


.98420 


13 


.17737 


.98414 


14 


.17766 


.93409 


15 


.17794 


.98404 


16 


.17823 


.98399 


17 


.17852 


.98394 


18 


.17883 


.93389 


19 


.17909 


.98383 


20 


.17937 


.93378 


21 


.17986 


.98373 


22 


.17995 


.98368 


23 


.18023 


.98362 


24 


.18052 


.93357 


25 


.18081 


.98352 


26 


.18109 


.98347 


27 


.18138 


.98341 


28 


.18166 


.98336 


29 


.18195 


.98331 


30 


.18224 


.98325 


31 


.18252 


.93320 


32 


.18281 


.98315 


33 


.18309 


.98310 


34 


.18338 


.93304 


35 


.1836? 


.93299 


36 


.18395 


.93294 


37 


.18424 


.98233 


38 


.18452 


.93233 


39 


.18481 


.93277 


40 


.18509 


.98272 


41 


.18538 


.98287 


42 


.18567 


.98231 


43 


.18595 


.98256 


44 


.18624 


.98250 


45 


.18632 


.98245 


46 


.18681 


.93240 


47 


.18710 


.93234 


43 


.18738 


.93229 


49 


.18767 


.93223 


50 


.18795 


.98218 


51 


.18321 


.98212 


52 


.18852 


.93207 


53 


.183^1 


.93201 


54 


.18910 


.98106 


55 


.18933 


.93190 


56 


.18967 


.98185 


57 


.18995 


.98179 


58 


.19024 


.98174 


59 


.19052 


.9816S 


60 


.19081 


.98163 


M. 


Cosin. | sine. 




* 


9° 



11° 


12° 


Sine. 

.19081 


Cosin. 


Sine. 


Cosin. 


.98163 


.20791 


.97815 


.19109 


.98157 


.20820 


.97809 


.19138 


.98152 


.20848 


.97803 


.19167 


.98146 


.20877 


.97797 


.19195 


.98140 


.20905 


.97791 


.19224 


.98135 


.20933 


.97784 


.19252 


.93129 


.20962 


.97778 


.19281 


.98124 


.20990 


.97772 


.19309 


.98118 


.21019 


.97766 


.19338 


.98112 


.21047 


.97760 


.19366 


.98107 


.21076 


.97754 


.19395 


.98101 


.2110? 


.97748 


.19423 


.93096 


.21132 


.97742 


.19452 


.98090 


.21161 


.97735 


.19481 


.98084 


.21189 


.97729 


.19509 


.98079 


.21218 


.97723 


.19538 


.98073 


.21246 


.97717 


.19566 


.98067 


.21275 


.97711 


.19595 


.98061 


.21303 


.97705 


.19623 


.93056 


.21331 


.97698 


.19652 


.98050 


.21360 


.97692 


.19680 


.98044 


.21388 


.97686 


.19709 


.98039 


.21417 


.97680 


.19737 


.98033 


.21445 


.97673 


.19768 


.98027 


.21474 


.97667 


.19794 


.98021 


.21502 


.97661 


.19823 


.98016 


.21530 


.97655 


.19851 


.98010 


.21559 


.97648 


.19830 


.98004 


.21587 


,97*542 


.19903 


.97998 


.21616 


.97636 


.19937 


.97992 


.21644 


.97630 


.19965 


.97987 


.21672 


.97623 


.19994 


.97981 


.21701 


.97617 


.20022 


.97975 


.21720 


.97611 


.20051 


.97969 


.21753 


.97604 


.20079 


.97963 


.21786 


.97598 


.20103 


.97958 


.21814 


.97592 


.20136 


.97952 


.21843 


.97585 


.20165 


.97946 


.21871 


.97579 


.20193 


.97940 


.21899 


.97573 


.20222 


.97934 


.21928 


.97566 


.20250 


.97928 


.21956 


.97500 


.20279 


.97922. 


.21985 


.97553 


.20307 


.97916 


.22013 


.97547 


.20336 


.97910 


.22041 


.97541 


.20364 


.97905 


.22070 


.97534 


.20393 


.97899 


.22098 


.97528 


.20421 


.97.-93 


. 22126 


.97521 


.20450 


.97887 


.22155 


.97515 


.20478 


.97881 


.22183 


.97508 


.20507 


.97875 


.22212 


.97502 


.20535 


.97869 


.22240 


.97498 


.20563 


.97863 


.22268 


.97489 


.20592 


.97357 


.22297 


.97483 


.20620 


-.97851 


.22325 


.97476 


.20649 


.91845 


.22353 


.97470 


.20677 


.97839 


.22382 


.97463 


.20706 


.97833 


.22410 


.97-157 


.20734 


.97827 


.22438 


.97450 


.20763 


.97821 


.22467 


.97444 


.20791 


.97815 


.22495 


.97437 


Cosin. | Sine. 


Cosin. | Sine. 


7 


8° 



13 

Sine. Cosin. 



.22495 
.22523 
.22552 

.£2580 
.22608 
.22687 
.22665 
.22693 
.22722 
.22750 
. 22778 
. 22807 
.228S5 
.22863 
.22892 
.22920 

.22948 

.22977 

.2300, 

.23033 

.23062 

, 23090 

.23118 

.23146 

.23175 

.23203 

.23231 

.23260 

.23288 

.23316 

.23345 

.23373 
.23-J01 
. 23429 

.23458 
.23486 
.23514 
. 23542 
.23571 
. 23599 
. 23627 
.23656 
. 236S4 
.23712 
. 23740 
.23769 

..23797 

.23825 
. ££853 
. 23882 
.23910 
.23938 
.28960 
.23995 
.24<23 
.24051 
.24079 
.24108 
.24136 
.24164 
.24192 



974£ 

97430 

97424 

.97417 

.97411 

.97404 

.97398 

.97391 

.97384 

.97378 

.9737 

.97365 

.97358 

.97351 

.97345 

.97338 

.97331 
.97325 
.97318 
.97311 
.97304 
.97298 
.97291 
.97284 
.97278 
.97271 
.97264 
.9725' 
.97251 
.97244 
.97237 

.97230 

.91228 

.9721 

.97210 

.91203 

.971 £6 

.971£9 

.97182 

.97176 

.97169 

.91162 

.97155 

.97148 

.97141 

.97134 

.97127 

.97120 

.9711 

.97106 

.97100 

.97093 

.97086 

,97079 

,97072 

,97065 

,97058 

97051 

97044 

97037 

97030 



14° 



Cosin. 



76° 



24192 
24220 
24249 
24277 
24305 
,24338 
,24362 
,24891 
.24418 
.24446 
.24474 
24503 
.24531 
.24559 
.2458' 
.24615 

.24644 
.24612 
.24700 
.24728 
.24756 
.24784 
.24818 
.24841 
.24869 
.24897 
.24925 
.24954 
.24982 
.25010 
.25038 

.25066 
.25(94 
.25122 
.25151 
.25179 
.25207 
.2523! 
.25203 
.85291 
1.25880 
L 25848 
1. 25870 
1.25404 
i.2t482 
.25460 

'.25488 
.25516 
1.25545 
.25573 
1.25601 
.25629 
.25657 
.256^5 
.25718 
.25741 
.25769 
.25798 
.25826 
.25854 
.25882 



97030 
97023 
97015 

97008 
97001 
96994 
.90987 
,90980 
,96973 
.96966 
.96959 
.96952 



.96923 

.96916 

.96909 
.96902 
.96894 
.96887 
.96880 
.96873 
.9fc66 
.96858 
.96851 
.96844 
.90837 
.96829 
.96822 
.90815 

.96807 
.96800 
.96793 
.96786 
.96' 
.96771 
.96764 
.96756 
.26749 
96742 
96734 
.96727 
96719 
96112 
96105 

96697 
.96090 
,90682 
96675 
96667 
96660 
.90653 
.90645 
.96038 
.90680 
.96623 
.90615 
.96608 
.96000 
.96593 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
46 

44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 

14 
13 
12 
11 
10 
9 



Cosin. | Sine. 

75° 



< 

( 
( 
I 
( 
( 

I 
U. ( 

( 



TABLE OP NATURAL SINES AND COSINES. 



159 





15° 


M. 


Sine. 


Cosin. 





.25882 


.96593 


1 


.25910 


.96585 


2 


.25938 


.96578 


3 


.25966 


.96570 


4 


.25994 


.96562 


5 


.26022 


.96555 


6 


.26050 


.96547 


7 


.26079 


.96540 


8 


.26107 


.96532 


9 


.26135 


.96524 


10 


.26163 


.96517 


11 


.26191 


.96509 


12 


.26219 


.96502 


13 


.26247 


.96494 


14 


.26275 


.96486 


15 


.26303 


.96479 


16 


£6331 


.96471 


17 


.26359 


.96463 


18 


.26387 


.96456 


19 


.26415 


.96448 


20 


.26443 


.96440 


21 


.26471 


.96433 


22 


.26500 


.96425 


23 


.26528 


.96417 


24 


.26556 


.96410 


25 


.26584 


.96402 


26 


.26612 


.96394 


27 


.26640 


.96386 


28 


.26668 


.96379 


29 


.26696 


.96371 


30 


.26724 


.96363 


31 


.26752 


.96355 


32 


.26780 


.96347 


33 


.26808 


.96340 


34 


.26836 


.96332 


35 


.26864 


.96324 


36 


.26892 


.96316 


37 


.26920 


.96308 


38 


.26948 


.96301 


39 


.26976 


.96293 


40 


.27004 


.96285 


41 


.27032 


.96277 


42 


.27060 


.96269 


43 


.27088 


.96261 


44 


.27116 


.96253 


45 


.27144 


.96246 


46 


.27172 


.96238 


47 


.27200 


.96230 


48 


.27228 


.96222 


49 


.27256 


.96214 


50 


.27284 


.96206 


51' 


.27312 


.96198 


52 


.27340 


.96190 


53 


.27368 


.96182 


54 


.273)6 


.96174 


55 


.27424 


.96166 


56 


.27452 


.96158 


57 


.27480 


.96150 


58 


.27508 


.96142 


59 


.27536 


.96134 


60 


.27564 


.96126 


M. 


Cosin. | Sine. 




7 


4° 



16° 



27564 
27592 
27620 
,27648 
,27676 
,27704 
27731 
27759 
,27787 
27815 
,27843 
,27871 
27899 
,27927 
27955 
,27983 

,28011 

28039 
,28067 
28095 
28123 

,28150 
,28178 
28206 
,28234 
,28262 
,28290 
,28318 
,28346 
,28374 
,28402 

,28429 

,28457 
,28485 
,28513 
,28541 
,28569 
,28597 
,28625 
,28652 
,28680 
,28708 
,28736 
,28764 
.28792 
.28820 

.28847 
.28875 



28931 
28959 
28987 
29015 
29042 
29070 
2909S 
29126 
29154 
29182 
29209 
29237 



Cosin. 



96126 
.96118 
96110 
96102 
.96094 
.96086 
.96078 
.96070 
.96062 
96054 
96046 
96037 
.96029 
.96021 
.96013 
.96005 

.95997 
.959S9 
.95981 
.95972 
.95964 
.95956 
.95948 
.95940 
.95931 
.95923 
.95915 
.9590' 
.95898 
.95S90 
.95882 

.95874 
.95365 
.9585' 
.95849 
.95841 
.95832 
.95824 
.95816 
.95807 
.95799 
.95791 
.95782 
.95774 
.95706 
.5975' 

.95749 
.95740 
.95732 
.95724 
.95715 
.9570' 
.95698 
.95690 
.95681 
.95673 
.95664 
.95656 
.9564' 
.95639 
.95630 



Cosin. | Sine. 



73° 



17° 



.29237 
.29265 
.29293 
.29321 

.29348 
.29376 
.29404 
.29432 
.29460 
.29487 
.29515 
.29543 
.29571 
.29599 
.29626 
.29654 

.29682 
.29710 
.29737 
.29765 
.29793 
.29821 
.29849 
.29876 
.29904 
.29932 
.29960 
.299S7 
.30015 
.30043 
30071 

.30098 
.30126 
.30154 
.30182 
.30209 
.30237 
.30265 
.30292 
:0320 
.30348 
.3'376 
.30403 
.30431 
.30459 
.30486 

.30514 
.30542 
.30570 
.30597 
.80625 
.30-653 
.30680 
.30708 
.30736 
.30763 
.30791 
.30819 
.30846 
.30874 
.30902 



.95630 
95622 
.95613 
.95605 
95596 
95588 
.95579 
95571 
.95562 
.95554 
.95545 
.95536 
.95528 
.95519 
.95511 
.95502 

.95493 

.95485 
.95476 
.95467 
.95459 
.95450 
.95441 
.95433 
.95424 
.95415 
.9540' 
.95398 
.95389 
.95380 
.95372 

.95363 
.95354 
.95345 
.9533 

.95328 
.95319 
,.95310 
.95301 
.95293 
.95284 
.9527; 
.95266 
.95257 
.95248 
.95240 

.95231 

.95222 
.95213 
.95204 
.95195 
.95186 
.95177 
.95168 
.95159 
.95150 
.95142 
.95133 
.95124 
.95115 
.95106 



18° 



19° 



SiEe. ) Cosin. Sine. Cot-in. 



Cosin. | Sine. 



30902 
30929 
30957 
30985 
31012 
.31040 
31068 
31095 
,31123 
.31151 
,31178 
.31206 
.31233 
.31261 
.31289 
.31316 

.31344 

.31372 
.31399 
.31427 
.31454 
.31482 
.31510 
.31537 
.31565 
.31593 
.31620 
.31648 
.31675 
.31703 
.31730 

.31758 
.31786 
.31813 
.31841 
.31868 
.31896 
.31923 
.31951 
.31979 
.32C06 
.32034 
.3 061 
.32C89 
.32116 
.32144 

.32171 
.32199 
.3222 r 
.32254 
.82282 
.32309 
.82337 
.32364 
,32392 
, 3?419 
.32447 
,32474 
,32502 
32529 
32557 



95106 
95097 
95088 
95079 
95070 
,95061 
95052 
95043 
,95033 
,95024 
,9roi5 
.96006 
.94997 



94970 

94961 
94952 
94943 
94933 
,94924 
,94915 



,94897 

,94888 

.948' 

.94869 

,94860 

.94851 

.94842 

,94832 

.94823 

.94814 

.94805 

,94795 

,94786 

,94777 

.94768 

,94758 

,94749 

,94740 

,94' 

,14721 

,9471 

,94702 

94693 

.94684 
.94674 

94665 
.94(56 
.94346 
.94687 

94627 
.94618 

94609 

94599 
.94590 

94580 

94571 
.94561 

94552 



.32557 
.32584 
.32612 
.32639 
.32667 
.32694 
.32722 
.32749 
1.32777 



Cosin. | Sine. Cosin. | Mne. 



.32832 

.32859 
.32887 
.32914 
32942 
32969 

32997 
,33024 
33051 
33079 
33106 
,33134 
,33161 
,33189 
,33216 
,38244 
,33271 
,38298 
,33326 
,3385, 
,33381 

,33408 
,83436 
,83463 
,33490 
,33518 
,33545 
,3857 
,386(0 
,3862' 
,38655 



.33710 
.33737 
.33' 
.33792 

.33819 

.38S*6 
.33874 
.38901 
.33929 
,38956 
,33983 
,34011 
,34038 
34065 
34093 
34120 
34147 
34175 
34202 



94552 
94542 
94533 
94523 
94514 
,94504 
,94495 
,94485 
,94476 
,94466 
.94457 
.94447 



94428 
94418 
94409 

94399 
94390 



94870 



94351 
94342 
94332 

94322 
,94313 



.94293 
.94284 
.94274 
.94264 

.94254 

.94245 
.94235 
.94225 
.94215 
.94206 
.94196 
.94186 
.94176 
.94167 
.94157 
.94147 
.94137 
.94127 
.94118 

,94108 

,94098 

,94088 

,94078 

,94( 

,94058 

94049 

94039 

94029 

94019 

94009 

93999 



3979 



70° 



60 
59 

58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 

44 
43 
42 
41 
40 



37 
36 
35 
34 
33 
32 
31 
30 

29 

28 
27 
26 
25 
24 
23 
22 
21 
20 
19 

is 

17 

16 
15 

14 
13 
12 
11 
10 



160 



TABLE OF NATURAL SINES AND COSINES. 





20° 


21° 


22° 


M. 


Sine. 


Cosin. 
.93969 


Sine. 
.35837 


Cosin. 


Sine. 
.37461 


Cosin. 





.34202 


.93358 


.92718 : 


1 


.34229 


.93959 


.35864 


.9334S 


.37488 


.92707; 


2 


.3425? 


.93949 


.35891 


.93337 


.37515 


.92697; 


3 


.34284 


.93939 


.35918 


.93327 


.37542 


.92680 


4 


.34311 


.93929 


.35945 


.93316 


.37569 


.92675; 


5 


.34339 


.93919 


35973 


.93306 


.37595 


.92664' 


6 


.34366 


.93909 


.36000 


.93295 


.37622 


.92653 


7 


.34393 


.93899 


.36027 


.93285 


.37649 


.9:642; 


8 


.34421 


.93389 


.36054 


.93274 


.37676 


.92631! 


9 


.34448 


.93879 


.36081 


.93264 


.37703 


. 92620! 


10 


.34475 


.93S69 


.36108 


.93253 


.37730 


.92609; 


11 


.34503 


.93859 


.36135 


.93243 


.37757 


. 92598 


12 


.34530 


.93349 


.36162 


.93232 


.37784 


.92587! 


13 


.34557 


.93839 


.36190 


.93222 


.37811 


.92576! 


14 


.34584 


.93829 


.36217 


.93211 


.37838 


. 92565 


15 


.34612 


.93819 


.36244 


.93201 


.37865 


.92554 


16 


.34639 


.93809 


.36271 


.93190 


.37892 


.92543, 


17 


.34666 


.93799 


.36298 


.93180 


. 37919 


. 92532 


18 


.34694 


.93789 


.36325 


.93169 


.37946 


. 92521 


19 


.34721 


.93779 


.36352 


.93159 


.37973 


. 92510 


20 


.34748 


.93769 


.36379 


.93148 


. 3799.) 


. 92499 


21 


.34775 


.93759 


.36406 


.93137 


. 38026 


. 92488 


22 


.34803 


.93748 


.36434 


.93127 


.38053 


. 92477 


23 


.34830 


.93738 


.36461 


.93116 


. 38080 


. 92466 


24 


.34857 


.93728 


.36483 


.93106 


. 38107 


. 92455 


25 


.34884 


.93718 


.36515 


.93095 


. 33134 


. 92444 


26 


.34912 


.93708 


.36542 


.93081 


.38161 


. 92432 


27 


.34939 


.93698 


.36569 


.93074 


. 38183 


.92421 


28 


.34966 


.93683 


.36596 


.93063 


.38215 


.92410 


29 


.34993 


.93677 


.36623 


.93052 


.38241 


. 92399 


30 


.35021 


.93667 


.36650 


.93042 


.38268 


. 92388 


31 


.35048 


.93657 


.36677 


.93031 


. 38295 


. 92377 


32 


.35075 


.93647 


.36704 


.93020 


. 38322 


. 92366 


33 


.35102 


.93637 


.95731 


.93010 


.38349 


. 92355 


34 


.35130 


.93626 


.36758 


.92999 


.38376 


. 92343 


35 


.35157 


.93616 


.36785 


.92988 


. 38403 


. 92332 


36 


.35184 


.93606 


.36812 


.92978 


.38430 


. 92321 


37 


.35211 


.93596 


.36839 


.92967 


.38456 


. 92310 


38 


.35239 


.93585 


.36867 


.92956 


.38483 


.92299 


39 


.35266 


.93575, 


.36S94 


.92945 


.38510 


.92287 


40 


.35293 


.93565! 


.36921 


.92935 


.33537 


. 92276 


41 


.35320 


.93555: 


.36948 


.92924 


.38564 


. 92265 


42 


.35347 


.93544; 


.36975 


.92913 


.38591 


. 92254 


43 


.35375 


.93534 


.37002 


.92902 


.38617 


.92243 


44 


.35402 


.93524! 


.37029 


.92892 


.38644 


. 92231 


45 


.35429 


.93514J 


.37056 


.92881 


.38671 


.92220 


46 


.35456 


.93503 


.37083 


.92870 


. 38698 


.92209 


47 


.35484 


.93493 


.37110 


.92859 


.33725 


.92198 


48 


.35511 


.93483 


.37137 


.92849 


.38752 


.92186 


49 


.35538 


.93472 


.37164 


.92338 


.33778 


.92175 


50 


.35565 


.93462 


.37191 


.92827 


.38805 


.92164 


51 


.35592 


.93452 


.37218 


.92816 


.38832 


.92152 


52 


.35619 


.93441 


.37245 


.92805 


.38859 


.92141 


53 


.35647 


.93431 


.37272 


.92794 


.38886 


.92130 


54 


.35674 


.93420 


.37299 


.92784 


.38912 


.92119 


55 


.357)1 


.93410 


.37326 


.92773 


.38939 


.92107 


56 


.35728 


.93400 


.37353 


.92762 


.38966 


.92096 


57 


.35755 


.93339 


.37380 


.92751 


.38993 


.92085 


58 


.35782 


.93379 


.37407 


.92740 


.39020 


.92073 


59 


.35810 


.93368 


.37434 


.92729 


.39046 


.92062 


60 


.35837 


.93358 


.37461 


.92718 


.39073 


.92050 


M. 


Cosin. | Sine. 

69° 


Cosin. | Sine. 


Cosin. | Sine. 

67° 




6 


8° 



23° 



39073 
,39100 
39127 
, 39153 
.391 SO 
.39207 
.39234 
. 39260 
. 89287 
.39314 
.39341 
. 39367 
.39394 
. 39421 
.39448 
. 39474 

.39501 

.89528 
. 89555 
.£9581 
.89608 
. 39635 
.39661 
.39688 
.89715 
.39741 
. 89768 
.89795 
.89822 
.39848 
.39875 

.89902 
. 89928 
. 39955 
.89982 
.40008 
.40085 
. 40C62 
. 40C88 
.40115 
.40141 
.40168 
.40195 
.40221 
.40248 
.40275 

.40301 
,40328 
, 40355 
40381 
,40408 
,40431 
40461 
4048S 
40514 
40541 
40567 
40594 
40621 
40617 
40674 



.92050 
92039 
92028; 
92016 
,92005 
91994 
91982 
.91971 
.91959 
.91948 
.91936 
.91925 
.91914 
.91902 
.91891 
.91879 

.91868 
.91856 
.91845 
.91833 
.9182: 
.91810 
.91799 
.9178* 
.91775 
.91764 
.91752 
.91741 
.91729 
.91718 
.91706 

.91694 
.91683 
.91671 
.91660 
.91648 
.91636 
.91625 
.91613 
.91601 
.91590 
.91578 
.91566 
.91555 
.91543 
.91531 

.91519 

.91508 
.91496 
.91484 
.91472 
,91461 
91449 
91437 
91425 
91414 
91402 
91390 
91378 
91366 

91355 

Cosin. | Sine. 

66^ I 



24° 



.40674 
.40700 
.40727 
.40753 
.40780 



.40833 
.40860 
.40F86 
.40913 
.40939 
.40966 
.40992 
.41019! 
.41045 
.41072 

.41098 
.41125 
.41151 
.41178 
1.41204 
1.41231 
.41257 
.41284 
.41310 
.41337 
.41363 
.41390 
.41416 
.41443 
.41469 

.41496 

.41522 

.41549 

.41575 

.41602 

.41628 

.41655 

.41681 

.41707 

.41784 

.41760 

.4178' 

.41813 

.41840 

.41866 

.41892 
.41919 
.41945 
.41972 
,41998 
,42024 
,42051 
42077 
42104 
42130 
42156 
42183 
42209 
42235 
42262 



.91355 
.91343' 
.91331 
.91319 
.91307 
.91295 
.91283 
.91272 
.91260 
.91*48 
.91236 
.91224 
.91212 
.91200 
.91188 
.91170 

.91164 
.91152 
.91140 
.91128 
.91116 
.91104 
.91092 
.91080 
.91068 
.91056 
.91044 
.91032 
.91020 
.91008 
.90996 

.90984 
.90972 
.90960 
.90948 
.90936 
.90924 
.90911 
.90899 
.90887 
.90875 
.90863 
.90851 
90839 
90821 
.90814 

.90802 
90790 
90778 

.90766 
90753 
90741 

.90729 
90717 
90704 
90692 

.90080 

,90668 
90655 
90643 

.90631 



Cosin. | Sine. 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 

44 
43 
42 
41 

40 



37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 





TABLE OF NATURAL SINES AND COSINES. 



161 



} M. 



8 
9 
10 
11 
12 
13 
14 
15 

16 
1? 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 

) 44 

) 45 
> 46 

)■ m 

\ 43 

I 49 

? 50 

( 51 

< 52 
53 
54 

S 55 

) 56 

) 57 

) 58 

) 59 

I 60 

L 



25° 



42262! 
,42288 
,42315 
4234H 
,42367| 
,42394! 
42420 
42416 
42473! 
42499 ' 
42525 
42552 ! 
42578 ' 
.43604! 
42631 
42657 ! , 

42683 
42709 
42736 
,42762 

42788 
42815 
42841 
42867 
42391 
42920 
42946 
42972 
42999 
43025 
43051 

43077 

43104 
431301 
43156! 
43182! 
43209 ( 
43235 
43261 
,43287 
,43313 
,43340 
,43366 
,43392 
,43418 
.43445 

.43471 
.43497 
.43523 
.43549 
.43575 
.43602 
.43628 
.43654 



90631 

90618 ! 
90606 

90594 
90582 
90569 
90557 
90545 
90532 
90520 ; 
90507! 
90495 
90483; 
90470 
90458 
90446 



26° 

Sine. Cosin^j 

89879!]" 
89867 

8! 1854 
89841 
89823 
89816 



~27° 



43706 
43733 
,43759 
,43785 
.43811 
.43837 



,90433 
,90421: 
,90408 
,90396' 
.90383 
,90371 
,90358 
,90346 
.90334 
.90321 
,90309 
.90298 
,90284,' 
.90271 
.90259: 

,90246 ! 
,90233: 
.90221; 

.90208: 
.90196; 
.90183! 
.90171! 
.901581 
.90146 
90133 
.90120 
90103 
90095 
90082; 
99070J 

.90057! 
90045! 
90032; 
.90019! 
.90007! 
.89994! 
.89931! 
.89963; 
.89956 ! 
.89943, 
.89930! 
.89918! 
.89905 1 ; 
.S9S92: 
.89879 



4383? 
43863 
43880 
43916 
43942 
43968 
43994 
44020 
44046 
44072 
44098 
44124 
j.44151 
1.44177 
1.44203 
.44229 

.44255 

.44281 
.44307 
i. 44333 
1.44359 
j.44385 
1.44411 
j. 44437 
1.44464 
1.44490 
.44516 
i. 44542 
.44568 
1.54594 
44620 

44646 
44672 

44698 
44724 
44750 
44776 
44802 
44828 
44354 
44880 
44906 
44932 
,44958 
,44984 
45010 

,45036 

,45062 

,451 

,45114 

.45140 

.45166 

.45192 

.45218 

.45243 

.45269 

.45295 

.45321 

.4534' 

.45373 

.45399 



Cosin. | Sine. | 



64° 



89790 
89777; 
89764 ! 
89752: 
89739! 
89726! 
89713 
89700! 
89637! 

89674 

89662 
89649 
89636 
.89623 
89610 
89597 
89534 
89571 
.89558 
89545 
89532 
89519 
89506 
.89493 

89480 

.89467 
89454 

.89441 
89423 

.89415 
89402 

.89389 
89376 
89363 
89350 
89337 
89324 
8931 
89298 

89285 
89272 
89259 
89245 
89232 
89219 



,98193 
,89180 
,89167 
.89153 
.89140 
.89127 
.89114 
.89101 



63° 



Sine. 

45399 
45425 
45451 
45477 
45503 
45529 
45554 
45580 
45606 
45632 
45658 
45084 
45710 
45736 
45762 
4578-J 

45813 
45839 
45865 
45891 
45917 
45942 
45968 
45994 
46020 
48046 
46072 
46097 
46123 
46149 
46175 

46201 
46226 
46252 
46278 
46304 
46330 
46355 
46381 
46407 
46433 
46458 
46484 
46510 
46536 
46561 

46587 
46613 
46639 
46664 
46690 
.46716 
.46742 
,46767 
.46793 
.46819 
.46844 
.46870 
.46896 
.46921 
.46947 



Cosi 



89101 
89087 
89074 
89061 
S9048 
89035 
8902: 
89008 
88995 



88968 

88955 
88942 
88928 
88915 
,88902 



88862 



88835 

.88S22 



88795 
88782 
88768 
88755 
88741 
88728 
88715 
88701 

88688 



88661 

88647 
88634 
88620 
88607 
8859? 
88580 
88566 
88553 
88539 
8852G 
88512 
88499 

88485 

88472 
88458 
88445 
88431 
8841' 
88404 
88390 
88377 
88363 
88349 
88336 
88322 



88295 



Cosin. | Sine. 

~62 



28° 


Sine. I 


Cosin 


.46947! 


^88295 


.46973 


.88281 


.46999! 


.8826? 


.47024 


.88254 


.47050 


.88240 


.47076 


.88226 


.47101 


.88213 


.47127 


.88199 


.47153 


.88185 


.47178 


.88172 


.47204 


.88158 


.47229 


.88144 


.47255 


.88130 


.47281 


.88117 


.47306 


.88103 


.47332 


.88089 


.47358 


.88075 


.47383 


.88062 


.47409 


.88048 


.47434 


.88(34 


.47460 


.88020 


.47486 


.88006 


.47511 


.87993 


.47537 


.87979 


a: 562 


.87965 


.47588 


.87951 


.47614 


.87937 


.47639 


.87923 


.47665 


.87909 


.47690 


.87896 


.47716 


.87882 


.47741 


.87868 


.47767 


.87854 


.47793 


.87840 


.47818 


.87826 


.47844 


.87812 


.47869 


.87798 


.47895 


.87784 


.47920 


.87770 


.47946 


.87756 


.47971 


.87743 


.47997 


.87729 


.48022 


.87715 


.48048 


.67701 


.48073 


.87687 


.48099 


.87673 


.4S124 


.87659 


.48150 


.87645 


.48175 


.87631 


.48201 


.87617 


.48226 


.87603 


.48252 


.87589 


.48277 


.87575 


.48303 


.87561 


.48328 


.87546 


.48354 


.87532 


.48379 


.87518 


.48405 


.87504 


.48430 


.87490 


.48456 


.87476 


.48481 


.87462 



29° 



Sire. 



Cosin. | Sine. 



61° 



48481 
48506 
48532 
4855? 
,48533 
,48608 
.48634 
.48659 
.48684 
.48710 
.48735 
.48761 
.48786 
.48811 
.48837 
.48862 

.48888 
.48913 
.48938 
.48964 
.48989 
.49014 
.49C4C 
.49065 
.49090 
.49116 
.49141 
.49166 
.49192 
.4921? 
.49242 



,49318 
,49344 
.49369 
49394 
49419 
4944 
49470 
49495 
.49521 
.49546 
.49571 
.49596 
.4962! 

.4964 
.49672 

|.4969 r 

.49721 
.49748 
.49773 
.49798 
1.49824 



.49874 
1.49899 
|. 49924 
.49950 
.49975 
.50000 



Cosin. 



87462 

87448 
87434 
87420 
.87406 
.87391 
87377 
87S63 
.87349 
.87335 
.87321 
.87306 
.87292 
.87278 
.37264 
.37250 

.87235 

.87221 
.87207 
.87193 
.87178 
.87164 
.87150 
.87136 
.87121 
.87167 
.87093 
.87079 
.67064 
.87050 
.87036 

.87021 
.87007 
.86993 
.£6978 
.66964 
.86949 
.86935 
.86921 
.86906 
.86892 
S6878 
.86863 
,86849 
86834 
86820 

86805 
86791 
86777 
86762 
86748 
86733 
86719 
86704 
86690 
86675 
86661 
66646 



86617 
86603 



sin. | Sine. 



60° 



60 ; 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 

44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 

28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15- 

14 
13 
12 
11 

10 
9 



5 
4 
3 
2 
1 

M s 



162 



TABLE OF NATURAL SINES AND COSINES. 



c*~ 


~^3oS~^ 


*~vg£~ 


~~32^" 


-— 33^— 


"2^34^ \ 


> M. 


Sine. 


Co in. 

.86603 


Sine. 

.51504 


Cosin. 

.85717 


Sine. 
.52992 


Cosi n . 


Sine. 


Cosin 


Sine. 


Cos'n. j 


s ° 


.50000 


.84305 


.54464 


.83867 


.55919 


.82904 


< 1 


.50025 


.86588 


.51529 


.85702 


.53017 


.84789 


.54488 


.83851 


.55943 


.82887 


I i 


.50050 


.81)573 


.51554 


.85687 


.53041 


.84774 


.54513 


.83835 


.55968 


.82871; 


.501)76 


.86559 


.51579 


.85672 


.53066 


.84759 


.54537 


.83819 


.55992 


.82855; 


? 6 


.50101 


.H0541 


.51604 


.85657 


.5301)1 


.84743 


.54561 


.83804 


.56016 


.82839 


.50126 


.86530 


.51628 


.85612 


.53115 


.84728 


.54586 


.83788 


.56040 


.82822 


.50151 


.86515 


.51653 


.85627 


.53140 


.84712 


.54610 


.83772 


.56064 


.8*806 


) 7 


.50176 


.86501 


.51678 


.85612 


.53164 


.8469? 


.54635 


.83756 


.56088 


.82790 


( 8 


.50201 


.86186 


.51703 


.85597 


.53189 


.84681 


.54659 


.83740 


.56112 


.82773 


> 9 


.50227 


.86171 


.51728 


.85582 


.53214 


.84666 


.54683 


.83724 


.56136 


.82757 


S 10 


.50252 


.86157 


.51753 


.85567 


.53238 


.84650 


.54708 


.83708 


.56160 


.82741 


I 11 


.50277 


.86442 


.51778 


.85551 


.53263 


.84635 


.54782 


.83692 


.56184 


.82724 


( 12 


.50302 


.86127 


.51803 


.85536 


.53283 


.84619 


.54756 


.83676 


.56208 


.82708 


) I 3 


.50327 


.86413 


.51823 


.85521 


.53312 


.84604 


.54781 


.88660 


.56232 


.82692 


> I 4 


.50352 


.86:398 


.51852 


.85506 


.53337 


.8-4588 


.54805 


.83645 


.56256 


.82675 


S 15 


.50377 


.86384 


.51877 


.85491 


.53361 


.84573 


.54829 


.83629 


.56280 


.82659 


S 16 


.50403 


.86369 


.51902 


.85476 


.53380 


.84557 


.54854 


.83613 


.56305 


.82643 


( 17 


.50128 


.86351 


.51927 


.85161 


.53411 


.84542 


.54878 


.83597 


.56329 


.82626 


) 18 


.50453 


.86310 


.51952 


.85446 


.53435 


.84526 


.54902 


.83581 


.56353 


.82610 


S 19 


.50178 


.86325 


.51977 


.85431 


.53460 


.84511 


.54927 


.83565 


.56377 


.82593 


) 20 


.50503 


.86310 


.52002 


.85416 


.53484 


.84495 


.54951 


.83549 


.56401 


.82577 


21 


.50528 


.86295 


.52026 


.85401 


.53509 


.84480 


.54975 


.83533 


.56425 


.82561 


22 


.50553 


.86281 


.52051 


.85385 


.53534- 


.84464 


.54999 


.83517 


.56449 


.82544 


23 


.50578 


.86266 


.52076 


.85370 


.53553 


.84448 


.55024 


.83501 


.56473 


.82528 


) 24 


.50603 


.86251 


.52101 


.85355 


.535S3 


.84433 


.55048 


.83485 


.56497 


.82511 


J 25 


.50628 


.86237 


.52126 


.85340 


,53607 


.84417 


.55072 


.83469 


.56521 


.82495 


J 26 


.50654 


.86222 


.52151 


.85325 


.53632 


.84402 


.55097 


.83453 


.56545 


.82478 


27 


.506; 9 


.86207 


.52175 


.85310 


.53656 


.84386 


.55121 


.83437 


.56569 


.82462 


28 


.50701 


.86192 


.522)0 


85294 


.53681 


.84370 


.55145 


.83421 


.56593 


.82446 


29 


.50729 


.86178 


.52225 


.85279 


.53705 


.84355 


.55169 


.83405 


.56617 


.82429 


f 30 


.50754 


.S6163 


.52250 


.85261 


.53730 


.84339 


.55194 


.83389 


.56641 


.82413 


1 31 


.50779 


.8614S 


.52275 


.85249 


.53754 


.84324 


.55218 


.83373 


.566(15 


.82396 


32 


.50804 


.86133 


.52299 


.85234 1 


.53779 


.84308 


.55242 


.83356 


.56689 


.82380 


33 


.50829 


.86119 


.52324 


.85218 


.53804 


.84292 


.55266 


.83340 


.56713 


.82363 


34 


.50851 


.86104 


.5234 ) 


.85203 


.53828 


.84277 


.55291 


.F3324 


.56736 


.82347 


35 


.50879 


.86089 


.52371 


.85188 


.53353 


.84261 


.55315 


.83308 


.56760 


.82330 


36 


.50904 


.86074 


.52399 


.85173 


.53877 


.84245 


.55339 


.83292 


.56784 


.82314 


) 37 


.50929 


.86059 


.52123 


.85157 


.53902 


.84230 


.55363 


.83276 


.56803 


.82297 


) 38 


.50954 


.86015 


.52448 


.85142 


.53926 


.84214 


.55388 


.83260 


.56832 


.82281 


) 39 


.50979 


.86030 


.52473 


.85127 


.53951 


.84198 


.55412 


.83244 


.56856 


.82264 


) 40 


.51001 


.86015! 


.52493 


.85112 


.53975 


.84182 


.55486 


.83228 


.5b880 


.82248 


> 41 


.51029 


.86000! 


.52522 


.85096 


.54000 


.84167 


.55460 


.83212 


.56904 


.82231 


) 42 


.51054 


.85935: 


.52547 


.85081 


.54021 


.84151 


.55484 


.83195 


.56928 


.82214 


( 43 


.51079 


.85970 


.52572 


.85066 


.54049 


.84135 


.55509 


.83179 


1.56952 


.82198 


( 44 


.51101 


.85956 


.52597 


.85051 


.54073 


.81120 


.55533 


.83163 


1.56976 


.82181 


I 45 


.51129 


.85911 


.52621 


.85035 


.54097 


.84104 


.55557 


.83147 


.67000 


.82165 


} 46 


.51154 


.85926' 


.52646 


.S5020 


.54122 


.84088 


.55581 


.83131 


.57024 


.82148 


> 47 


.51179 


.85911 


.52671 


.85005 


.54146 


.84072 


.55605 


.83115 


.57047 


.82132 


( 48 


.51201 


.85896 


.52696 


.81989 


.51171 


.84057 


.55630 


.83098 


.57071 


.82115 


? 49 


.51229 


.85881! 


.52720 


.84974 


.54195 


.84041 


.55654 


.83182 


.57095 


.82098 


( 50 


.51254 


.85866; 


.52745 


.81959 


.54220 


.84025 


.55678 


.8: 1066 


.57119 


.82082 


< 51 


.51279 


.85851! 


.52770 


.84943 


.54244 


.84009 


.55702 


.83050 


.57143 


.82065 


( 52 


.51301 


.85836 


.52794 


.84928 


.54269 


.83994 


.55726 


.83034' 


.57167 


.82048 


< 53 


.51329 


.85821; 


.52819 


.84913 


.542D3 


.83978 


.55750 


.83017 


.57191 


.82032 


S 51 


.51354 


.85806 , 


.52844 


.84897 


.54317 


.83962 


.55775 


.83001 


.57215 


.82015 


> 55 


.51379 


. 8579-2 , 


.52869 


.84882 


.54342 


.83946 


.55799 


.82985 


.57238 


.81999 


> 56 


.51404 


.85777 


.52893 


.84866 


.54366 


.83930 


.55823 


.82969 


.57262 


.81982 


> 57 


.51429 


.8)762 


.52918 


.84851 


.54391 


.83915 


.55847 


.82953 


.57286 


.81965 


) 58 


.51451 


.85747 


.52943 


.84836 


.54415 


.83899 


.55871 


.82936 1.57310 


.81949 


> 59 


.51479 


.85732 


.52967 


84820 


.54440 


.83883 


.55895 


.82920 


.57334 


.81932 


? 60 

J M. 


.51504 


.85717 


.52992 


.84805 


.54464 


.83867 


.55919 


.82904 


1.57358 


.81915 


Cosin. | Sine. 

TiF" 


Cosin. | Sine. 


Cosin. | Sine. 

I '57°" 


Cosin. | Sine. 

56° 


j Cosin. | ^ine. 

55° 


5 


8° 



TABLE OF NATURAL SINES AND COSINES. 



163 



2 




5° 


> M. 


Sine. 


Cosin. 


u 


.57358 


.81915 


.57381 


.81899 


I 2 


.57405 


.81882 


\ «* 


.57429 


.81865 


.57453 


.81848 


5 


.57477 


.81832 


J 6 


.57501 


.81815 


) 8 


.57524 


.81798 


.57548 


.81782 


< 9 


.57572 


.81765 


< 10 


.57596 


.81748 


s 11 


.57619 


.81731 


< 12 


.57643 


.81714 


S 13 

< 14 

15 
X 16 


.57667 


.81698 


.57691 


.81681 


.57715 


.81664 


.57738 


.81647 


< 17 


.57762 


.81631 


18 


.57786 


.81614 


> 19 


.57810 


.81597 


5 20 
;. 21 


.57833 


.81580 


.57857 


.81563 


; 22 


.57881 


.81546 


; 23 


.57904 


.81530 


I 24 


.57923 


.81513 


' 25 


.57952 


.81496 


- 26 


.57976 


.81479 


'•' 27 


.57999 


.81462 


'. 28 
': 29 


.53023 


.81445 


.58047 


.81428 


; 30 


.58070 


.81412 


< 31 


.58094 


.81395 


\ 32 


.58118 


.81378 


( 33 


.58141 


.81361 


< 34 


.58165 


.81344 


' 35 


.58189 


.81327 


< 36 


.58212 


.S1310 


) 37 


.58236 


.81293 


; 38 


.58260 


.81276 


;■ 39 


.58233 


.81259 


/ 40 


.58307 


.81242 


, 41 


.58330 


.81225 


> 42 


.58354 


.81208 


i 43 


.58378 


.81191 


44 


.58401 


.81174 


45 


.58425 


.81157 


' 46 


.58449 


.81140 


47 


.58472 


.81123 


• 48 


.58496 


.81106 


49 


.58519 


.81089 


' 50 


.58543 


.81072 


51 


.58567 


.81055 


52 


•58590 


.81033 


53 


.58614 


.81021 


54 


.58637 


.81001 


55 


.58661 


.80987 


> 56 


.536S4 


.80970 


} 57 


.58708 


.80953 


> 58 


.58731 


.80936 


> 59 


.58755 


.80919 


? 60 


.58779 


.80902 


S M. 


Cosin. | ^ine. 


5 


" iT 


4° 





2o 


S7° 


38° 


39° 


M. ] 


Sine. 


Cosin 

.80902 


Sine. 
.60182 


Cosin. 

.79864 


Sine. 


Co in. 

.78801 


Sine. 


Cosin. | 


.58779 


.61566 


.62932 


.77715 60 ) 


.58802 


.80885 


.60205 


.79816 


.61589 


.78783 


,62955 


.77696 


59 > 


.58826 


.80867 


.60228 


.79829 


.61612 


.78765 


.62977 


.77678 


58 ) 


.58849 


.80850 


.60251 


.79811 


.61635 


.78747 


.63000 


.77660 


57 ( 


.58873 


.80833 


.60274 


.79793 


.61958 


.78729 


.63022 


.77641 


56 < 


.53896 


.80816 


.60298 


.79776 


.61681 


.78711 


.63045 


.77623 


55 < 


.58920 


.80799 


.60321 


.79758 


.61704 


,78694 


.63068 


.77605 


54 ( 


.58943 


.80782 


.60344 


.79741 


.61726 


.78676 


.63090 


.77586 


53 


.58967 


.80765 


.60367 


.79723 


.61749 


.78658 


.63113 


.77568 


52 


.58990 


.80748 


.60390 


.79706 


.61772 


.78640 


.63135 


.77550 


51 


.59014 


.80730 


.60414 


.79688 


.61795 


.78622 


.63158 


.77531 


50 < 


.59037 


.80713 


.60437 


.79671 


.61818 


.78604 


.63180 


.77513 


49 > 


.59061 


.80696 


.60460 


.79653 


.61841 


.78586 


.63203 


.77494 


48 > 


.59084 


.80679 


.60483 


.79635 


.61864 


.78568 


.63225 


.77476 


47 > 


.59108 


.80662 


.60506 


.79618 


.61887 


.78550 


.63248 


.77458 


46 ) 


.59131 


.80644 


.60529 


.79600 


.61909 


.78532 


.63271 


.77439 


45 ) 


.59154 


.80627 


.60553 


.79583 


.61932 


.78514 


.63293 


.77421 


44 > 


.59178 


.80610 


.60576 


.79565 


.61955 


.78496 


.63316 


.77402 


43 ) 


.59201 


.80593 


.60599 


.79547 


.61978 


.78478 


.63338 


.77384 


42 ; 


.59225 


.80576 


.60622 


.79530 


.62001 


.78160 


.63361 


.77366 


41 > 


.59243 


.S055S 


.60645 


.79512 


.62024 


.78442 


.63383 


.77347 


40 < 


.59272 


.80541 


.60668 


.79494 


.62046 


.78424 


.63406 


.77329 


39 < 


.59295 


.80524 


.60691 


.79477 


.62069 


.78405 


.63428 


.77310 


38 ( 


.59318 


.80507 


.60714 


.79459 


.€2092 


.78387 


.63451 


.77292 


37 ( 


.59342 


.80489 


.60738 


.79441 


.62115 


.78369 


.63473 


.77273 


36 ( 


.59365 


.80472 


.60761 


.79424 


.62138 


.78351 


.63496 


.77255 


35 S 


.59389 


.80455 


.60784 


.79406 


.62160 


.78333 


.63518 


.77236 


34 < 


.59412 


.80438 


.60807 


.79388 


.62183 


.78315 


.63540 


.77218 


33 ) 


.59436 


.80420 


.60330 


.79371 


.62206 


.78297 


.63563 


.77199 


32 > 


.59459 


.80403 


.60853 


.79353 


.62229 


.78279 


.63585 


.77181 


31 > 


.59482 


.80386 


.60876 


.79335 


.62251 


.78261 


.63608 


.77162 


30 > 


.59506 


.80368 


.60899 


.79318 


.62274 


.78243 


.63630 


.77144 


29 ] 


.59529 


.80351 


.60922 


.79300 


.62297 


.78225 


.63653 


.77125 


28 ) 


.59552 


.80334 


.60945 


.79282 


.62320 


.7b206 


.63675 


.77107 


27 S 


.59576 


.80316 


.60968 


.79264 


.62342 


.78188 


.63698 


.77088 


26 ; 


.59590 


.80299 


.60991 


.79247 


.€2365 


.78170 


.63720 


.77070 


25 ) 


.59622 


.80282 


.61015 


.79229 


.62388 


.78152 


.63742 


.77051 


24 ) 


.59646 


.80264 


.61038 


.79211 


.62411 


.78134 


.63765 


.77033 


23 I 


.59669 


.80247 


.61061 


.79193 


.62483 


.78116 


.63787 


.77014 


22 \ 


.59693 


.80230 


.61084 


.79176 


.62456 


.7809S 


.63810 


.76996 


21 I 


.59716 


.80212 


.61107 


.79158 


.62479 


.78079 


.63832 


.76977 


20 I 


.59739 


.80195 


.61130 


.79140 


.62502 


.78061 


.63854 


.76959 


19 < 


.59763 


.80178 


.61153 


.79122 


.62524 


.78043 


.63877 


.76940 


18 < 


.59786 


.80160 


.61176 


. 79105 


.62547 


.78025 


.63899 


.76921 


17 < 

16 

15 

14 

13 i 
12 5 


.598)9 


.80143 


.61199 


.79087 


.62570 


.78007 


.63922 


.76903 


.59832 


.80125 


.61222 


.79069 


.62592 


.77988 


.63944 


.76884 


.59856 


.80108 


.61245 


.79051 


.62615 


.77970 


.63966 


.76866 


.59879 


.80091 


.61268 


.79033 


.62638 


.77952 


.63989 


.16847 


.59902 


.80073 


.61291 


. 79016 


.62660 


.77934 


.64011 


.76328 


59926 


.80056 


.61314 


.78993 


.62683 


.77916 


.64033 


.76810 


11 

10 > 


.59949 


.80033 


.61337 


.78980 


.€2706 


.77897 


.64056 


.76791 


.59972 


.80021 


.61360 


.78962 


.62728 


.77879 


.64078 


.76772 


9 

8 


.59995 


.80003; 


.61383 


.78944 


.62751 


.77861 


.64100 


-76754 


.60019 


.79986 


.61406 


.78926 


.62774 


.77843 


.64123 


.76735 


7 ( 


.60042 


.79968 


.61429 


.78908! 


.62796 


.77824 


.64145 


.76717 


6 ( 


.60065 


.79951 


.61451 


.78891! 


.62819 


.77806 


.64167 


,76698 


5 ( 


.60089 


,79934 


.61474 


.78873 


.62842 


.77788 


.64190 


.76679 


4 < 


.60112 


.79916 


.61497 


.788551 


.62864 


.77769 


.64212 


.76661 


3 < 


.60135 


.79899 


.61520 


.78837 


.62887 


.77751 


.64234 


.76642 


2 ) 


.60158 


.79881 


.61543 


.78819 


.62909 


.77733 


.64256 


.76623 


1 < 


.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


64279 


.76604 


S 


Cosin. | Sine. 

53° 


Cosin. | Sine. 


Cosin. | Sine. 


Cosm. | Sine. 

50° 


" M. > 


5] 


L° 


I 



164 



TABLE OF NATURAL SIXES AND COSINES. 



1 


40° 


41° 


M. 


Sine. 
.64-279 


Ci>sin. 


Sine. 
.65606 


Cosin. 

.75471 





.76604 


1 


.6430! 


.76586 


.65628 


.75452 


2 


.61323 


.76567 


.65650 


.75433 


3 


.64346 


.66548 


.65672 


.75414 


4 


.643(38 


.76530 


.65694 


.75395 


5 


.64390 


.76511 


.65716 


.75375 


6 


.64412 


.76492 


.65738 


.75356! 


7 


.64435 


.76473 


.65759 


.75337 


8 


.64457 


.76455 


.65781 


.75318 


9 


.64479 


.76436 


.65803 


.75299 


10 


.64501 


.76417 


.65825 


.75280 


11 


.64524 


.76398 


.65847 


.75261 


12 


.64546 


.76380 


.65869 


.75241 


13 


.64568 


.76361 


.65891 


.75222 


14 


.64590 


.76342 


.65913 


.75203 


15 


.64612 


.76323 


.65935 


.75184 


16 


.64635 


.76304 


.65956 


.75165 


17 


.64657 


.76286 


.65973 


.75146 


18 


.64679 


.76267 


.66000 


.75126 


19 


.64701 


.76248 


.66022 


.75107 


20 


.64723 


,76229 


.66044 


.75088 


21 


.64746 


.76210 


.66066 


.75069 


22 


.64763 


.76192 


.660S8 


.75J50II 


23 


.64790 


.67173 


.66199 


.75030 


24 


.64812 


.76154 


.66131 


.750111 


25 


.64834 


.76135 


.66153 


.74992 


26 


.64856 


.76116 


.66175 


.74973 


27 


.64878 


.76097 


.661 )7 


.74953: 


28 


.64901 


.76078 


.69218 


.74934 


29 


.64923 


.76059 


.6624 ) 


.74915' 


30 


.64945 


.76041 


.66262 


.74896 


31 


.64967 


.76022 


.66281 


.74873' 


32 


.64)89 


.76003 


.66303 


.74857; 


33 


.65011 


.75984 


.66327 


.74838! 


34 


.65033 


.75965 


.66349 


.74818; 


35 


.65055 


.75949 


.66371 


.74799 


36 


.65077 


.75927 


.66393 


.74780! 


37 


.65109 


.75908 


.66414 


.747601 


38 


.65122 


.75889 


.66136 


.74741! 


39 


.65144 


.75870 


.66458 


.74722 


40 


.65166 


.75851 


.66480 


.74701: 


41 


.65188 


.75832 


.665 '1 


.746&3 


42 


.65210 


.75813 


.66523 


.74664; 


43 


.65232 


.75794 


.66545 


.74614 


44 


.65254 


.75775 


.66566 


.74625 


45 


.65276 


.75756 


.66588 


.74606 


46 


.65298 


.75738 


.66610 


.74586 


47 


.65320 


.75719 


.66632 


.74567 


48 


.65342 


.75700 


.66653 


.74548 


49 


.65364 


.75680 


.66675 


.74528 


50 


.65386 


.75661 


.66697 


.74509 


51 


.65408 


.75642 


.66718 


.74489' 


52 


.05430 


.75623 


.66740 


.74470: 


53 


.65452 


.75694 


.66762 


.74451 


54 


.65474 


.75585 


.66783 


. 74431 ; 


55 


.65496 


.75566 


.60805 


.74412 


56 


.65518 


.75547 


.66827 


.74392 


57 


.65540 


.75528 


.66848 


.74373 


58 


.65562 


.75509 


.66870 


.74353' 


59 


.65584 


.75490 


.66891 


.74334 


60 


.65606 


.75471 


! . 66913 


.74314 


M. 


Cosin. | Sine. 


Cosin. | Sine. | 




" 4 


9° 


4 


8° 



.66913 
.66935 
.66956 
.66978 
.66999 
.67021 
.67043 
.67064 
.67086 
.67107 
.67129 
.67151 
.67172 
.67194 
.67215 
.67237 

.67258 
.67280 
.6,301 
.67322 
.67344 
.67366 



.74120 
.74100 
.74080 
.74061 
.74041 
.74022 



73708 
736S8 
,73669 
.73649 
,73629 
,73611 
.73590 
,73570 
,73551 
,73531 
,73511 
,73491 
,73472 
,73452 
,73432 

.73413 
.73393 
.73373 
,73:353 
.73333 
,73314 
.73294 
.73274 
.73254 
.73234 
.73215 
.73195 
.73175 
.73155 
.73135 



43° 


Sine. 


Cos-in. 


.6*200 


. 13135 i 


.68221 


.73116 


.68242 


.73090; 


.68264 


.73076 


.68285 


.73056 


.6S306 


.73036 


.68327 


.73016 


.68349 


.72996 


.68370 


.72976; 


.68391 


.72957 


.68412 


.72937J 


.684:34 


.72917! 


.68455 


.72S97 : 


.68476 


.72877. 


.68497 


.72857: 


.08513 


.72831; 


'.€8539 


.72817* 


.08561 


.72197; 


.68582 


.72777; 


.68603 


,72757; 


.6-624 


.72737, 


.6 C 645 


.12717 


.68666 


.72097 


.68088 


.72677 


.68709 


.72657 


.68730 


.72037 


.08751 


.12617 


.68772 


.72597 


.08793 


.72577 


.68814 


.72557 


;.68835 


.7:2537 


1.68857 


.72517 


.08878 


.72497 


.05899 


.72477 


.68920 


.72451 


.08941 


.72437 


.08962 


.72411 


.68983 


.72397 


.69004 


.72377 


.69025 


.72357 


1.09046 


.72337 


j. 69007 


.72317 


.09088 


.72297 


I.091C9 


.72277 


.09130 


.72257 


.09151 


.72230 


.69712 


.72210 


.69193 


.72190 


.09214 


.72170 


.69235 


.72150 


.09256 


.72136 


.09277 


.72116 


.69298 


.72095 


.69319 


.72075 


.69340 


.72055 


.09361 


.72035 


.09382 


.72015 


.69403 


.71995 


.69424 


.71974 


.69445 


.71954 


.69466 


.71934 


Cosin. | Sine. 


4( 


5° 



~4^~r^ 



Sine. 

.69466 
.69487 
.69508 
.09529 
.09549 
.09570 
.C9591 
.09612 
.69633 
.69654 
.09075 
.€9696 
.(9717 
.0973: 
.0975s 
.09779 

.69S00 

.09821 J 
.09842J 
.09862! 
.09883! 
.09904! 
i. 09925 
j.09946 
i. 09966 
.09987 
.70008 
1.70029 
.70049 
.700. 
.70091 



Cosin. M. 



53 
52 
51 
50 
49 
48 
47 
46 



.70112 
.70132 
.70153 
.70174 
.10195 
.10215 
.70236 j 
.70257 
.70277 
.70298 
.70319 
.70339 
.70300 
.10881 
.10401 

.10422 

.10443: 
.10403, 
.704841 
.70505' 
.70525: 
.70546 
.10567! 
.10581! 
.70608 
.70628| 
.70649 
.70670! 
.70090! 
.70711J 



.71934 60 
.71914 59 
.71894; 58 
.71873 1 57 
.71853 56 
.71833 55 
.71813 54 
.71792 
.71772 
.71752 
.11732 
.71711 
.71691 
71671 
.71650 
.71630 

.71610 44 
.71590 43 
.71509 42 
.71549 41 
.71529' 40 
.71508 39 
.71488 38 
.71468 37 
.71447: 36 
.71427 35 
.71407; U 
.71397 33 
.71366J 82 
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THE NORMAL PUBLISHING HOUSE. 



Popular and Practical Publications for 
Teachers and Others. 



THE NORMAL QUESTION BOOK, 

Containing over 4,000 questions and answers on the common 
branches, English and American Literature, Civil Government 
and Parliamentary Law, with an appendix on Methods of Teach- 
ing, etc. JSil^JusT the Book to Prepare Teachers for an 
Examination. Price, $1.50, post-paid, to any address. 



Methods of Teaching in Country Schools. 

THE BEST WORK ON TEACHING. 

Gives the very best methods extant. No work to compare with 
it for the country teacher. If you would make your school a 
grand success order this work at once. Price, $1.25. 



Normal Outlines of the Common School 
Branches. 

This much needed work fills a want not met by any other work. 
It lifts the teacher out of the ruts and puts him on the highroad 
of success in learning and teaching the common branches. It is 
a development of the plan so successfully pursued in the Inde- 
pendent Normal Schools, and to which they owe their success in 
developing the thinking and reasoning faculties of their students. 
If you wish to learn the secret of fully mastering any subject, 
study this book. Price, $1.00. 



Outline of Elocution and Comprehensive Man- 
ual of Principles. 

This is a superior self-instructor in the Art of Elocution and 
Oratory. Do you wish to prepare yourself for a reader or speak- 
er? If so, by all means secure this book. A thorough mastery 
of the principles herein set forth will enable you to declaim or 
speak accurately and ably, provided you have any taste or aspira- 
tion in that line. In short, if you are preparing for a speaker 
you can not afford to do without this book. Price, $1.50. 



J^Kull Descriptive Catalogue of tnese and other 
Books upon application. Address 

J. E. SHERRILL, Prop'r, 

DANVILLE, IPSO. 



THE NORMAL PUBLISHING HOUSE 



Popular and Practical Publications for 
Teaclters and Others* 



DIAGRAMS AND ANALYSES, 

7£y that Prince of Grammarians, the late Prof. Frank P. Adams, 
Principal of the Central Normal College, Danville, Ind. This is 
a Key to Harvey's and Hoebrook's Grammars. It is a work 
fully up to the times, and one needed by every teacher and gram- 
marian. Price, $1.25. 

"aUEER OXJERIES," 

A collection of 1,000 queer questions upon a multitude of sub- 
jects. Price 25 cents. ANSWEBS to the same, 10 cents. 

The Normal Teacher Parsing Book and the 
Normal Diacritical and Blank Speller 

Have been specially prepared with the view of making these 
subjects interesting and fascinating to the pupils. All who have 
used them have been delighted with the progress of their classes. 
Price of each, 20 cents. Copies for examination, with a view to 
introduction, 15 cents each. Examine these books and you will adopt 
ihem. 



The Normal Speaker and the Normal 
Dialogue Book 

Will meet the wants of those preparing for exhibitions and en- 
tertainments. There is no better way of interesting your pupils 
and their parents in the school, than by giving an occasional en- 
tertainment. These books will be just what you need. Price 50 
cents each. Order the books, and get up an entertainment. 

Pleasant Songs for Pleasant Places 

Is a very popular singing book for schools, entertainments and 
the home circle. Eetail, 20 cents. For examination, with a view 
to adoption, 15 cents. $1.50 per dozen, for use in schools. 

Opening Exercises 

Solves the problem of "What shall I do to open school prop- 
erly ?" Sample, for examination, 15 cents. $1.50 per dozen. 

Full descriptive catalogue of these and other books upon 
application • Address 

J. E. SHERRILL, Proper, 

DANVILLE, IND. 



Popular and Practical Publications for 
Teachers and Others. 



OUTLINES OF XT. S. HISTORY. 

This book presents, at a glance, a bird's-eye view of the History 
of our Country, and gives the best methods of teaching the sub- 
ject. It is by all odds the best work of the kind published. 

Price, 75 qents. 

The New Method, or School Expositions, 

Gives the Normal system of teaching. It is from the pen of Prof, 
E. Heber Holbrook, of Lebanon, Ohio, and shows how they man- 
age their expositions, so popular and instructive, and gives ex- 
plicit directions as to how these expositions may be carried out iel 
any countrv school. It is an invaluable work to any teacher. 
Price, $L00. 

Normal History *of the United States. 

This is pronounced by all who have examined it to be the coming- 
history. It only needs to be examined to be adopted. Finely 
illustrated, with maps, engravings and plats. Eetail price, $1.35^ 
Sample for examination with a view to introduction, $1.00. 



Easy Experiments in Chemistry and Natural 
Philosophy. 

Can be used as a text book in any school, or will serve as a work 
for opening exercises. One of the most successful ways to use 
this book is to place copies in the hands of the larger pupils and 
have them perform the experiments at the opening of the school 
in the morning and Friday afternoons. The experiments can all 
be performed with apparatus manufactured by the pupils. During- 
one term of school a teacher can give his pupils a very good idea 
of these subjects by the use of this book. Price, 40 cents. 



Crosier's Digest of Infinitives and Participles, 
Abridgment and Abridged Propositions, 

Is a work of great merit. It fully explains these most difficult 
subjects. • Every teacher ought to have a copy. Only 40 cents. 

«®"Kull Descriptive Catalogue of these and other 
Books upon application. Address 

J. E. SHERRILL, Proper, 

DANVILLE, INO. 



"QUEER QUERIES." 

A BOOK FOR THE STUDENT. 

A BOOK FOR THE TEACHER. 

A BOOK FOR EVERYBODY. 

A COLLECTION OF QUESTION'S ON DIFFERENT BRANCHES OF 

STUDT. 

This system of teaching " thing's not in the books " has been in use in many 01 
the public schools for several years, and has met with almost unlimited success in 
being- the means of incucating facts and principles into the j'outhful mind which 
can hardly be impressed upon the memory in any other way. It will lead to inves- 
tigations and researches on the part of the student which cannot prove otherwise 
than beneficial. Creates great interest in schools, at Institutes, wherever used. 

PREFATORY and EXPLANATORY. 

Queer Queries were collected in the following manner, viz : pupils were requested 
to bring any query which they thought would interest others or which they could 
not answer themselves, to the teacher. 

1 he teacher then placed ten of the first queries found in this little book upon the 
black-board and allowed them to remain there from Monday morning till Friday 
evening, when they were answered in a general exercise in which all the pupils 
shared equallv. 

The result was that the school closed with a good understanding of why the time 
in China and America are not the same, of why the feet of the Chinese point tow- 
ard our own ; of why the sun seems to rise in the east, of why Patagonia has no 
Capital, &c. 

The time occupied in this work was not to exceed ten minutes. 

The teacher tried this experiment the next week with the succeeding te?i questions 
with the school thoroughly alive to this new departure: every question was intelli- 
gently discussed by the pupils, both old and young. 

The third week two or three heads of families sent queries (See Nos. 23, 27 and 
29), and the interestincreased. The teacher kept up this system with no visible in- 
dication of lagging interest for one himdred weeks with the very best results. 

The demand for queries has been so great that we have consented to publish our 
first one thousand " Queer Queries." 

How to use Qeer Queries? take the book on Friday evening and call the atten- 
tion of the school to such queries as you may have selected by having the pupils to 
mark them by numbers; thus if you think it not best foryour school to take them in 
regular order and you should, select Nos. 1, 2, 3, 4, 5, 7, 13, &c. ;— 
have, the pupils "check " those numbers telling them they may study 
the questions at odd times till the next Friday evening when they 
may see who can answer the greatest number out of the ten selected. 
Pupils will individually ask you during the week to answer certain questions which 
they fail to find satisfactory theory for. Cite them to text-books, authors or per- 
sons within your knowledge where they will probably obtain the desired informa- 
tion; iu no case should you give the desired information direct to the individual ; 
but should the school as a body not be able to answer a question satisfactorily, then 
will be the time to help it out of the dilemma by gradually and pleasantly leading 
the school to see and know the why and therefore of the subject under 
consideration. 

Object of Queer Queries: 

1 st. To lessen the care of the teacher and make the school more attractive for 
the pupils by adding spice to at least one exercise for the week. {The last day's 
work should be the most pleasaiif). 

2nd. To form habits of close observation in the growing pupil, and in forming 
these habits which will cling to him through life, give him a fund of information 
which will vjell repay for alt the trouble and time rvhich such a plan imposes. 

Order a supply at once for your school. Agents wanted. No trouble to sell 
this little book. Give it a trial and be convinced. Price, 25 cents; $1.60 per dozen, 
postpaid. Published by the Normal Publishing House, Danville, Ind. 



